V (vim), 105 bytes

"aDiaibgyor<esc>0vg_y$@ap0l@alhd$:s/a/ v/g
qqYP0xjYP!!rev
gJkhq2@a@qylpp02ld$0VG:center<c-r>=2*(<c-r>a+ceil(1+<c-r>a/7))

Try it online!

There's many possible saves here, but for now I'm just happy I got it working.

PowerShell, 108 98 89 85 bytes

param($x)($x+=$x/7-replace'\..*')..0|%{' '*$_+-join(" vibgyor"*$x)[$x..$_+$_+$_..$x]}

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This one feels pretty alright now. Banker's rounding is still the devil and I figured out how to make a non-dumb join. I tried monkeying with $ofs to not much success. Speaking of, the results without joins look pretty good, a bit melty:

         vvv
        v     v
       v   rrr   v
      v   r ooo r   v
     v   r o yyy o r   v
    v   r o y ggg y o r   v
   v   r o y g bbb g y o r   v
  v   r o y g b iii b g y o r   v
 v   r o y g b i vvv i b g y o r   v
v   r o y g b i v     v i b g y o r   v

Dyalog APL, 41 39 38 bytes

↑{⌽(⌽,⊃,A↑⊢)⍵↑A⍴' vibgyor'}¨-⍳A←⌈⎕×8÷7

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A similar approach to others: A←⌈⎕×8÷7 finds the height of the rainbow (also the width of the longest 'half row' to the left/right of the centre) and assigns it to A for later use, while ¨-⍳ iterates through the values 1..A, negating them to select on the correct side when used with ↑.

A⍴' vibgyor' generates a 'half row' and ⍵↑ selects the correct length substring. (⌽,⊃,A↑⊢) generates the full row in reverse (which takes fewer characters to do), starting with a reversed half row (⌽), then the centre character taken from the beginning of the half row string (⊃) and finally a right padded version of the half row (A↑⊢). The final ⌽ reverses the row into the correct orientation and ↑ turns the vector of rows into a 2D array.

Edit: -2 thanks to dzaima

Edit: -1 thanks to ngn

Stax, 23 bytes

⌡G'5h!M╩EV[Ez ▼>≈<S⌡⌡0`

Run and debug it

Unpacked, ungolfed, and commented, it looks like this.

" vibgyor"  string literal
,8*7/^      input * 8 / 7 + 1
:m          repeat literal to that length
|]          get all prefixes
Mr          rectangularize, transpose array of arrays, then reverse
            this is the same as rotating counter-clockwise
m           map over each row with the rest of the program, then implicitly output
            the stack starts with just the row itself
  _h        push the first character of the row
  _r        push the reversed row
  L         wrap the entire stack in a single array

Run this one

Haskell, 106 113 bytes

I can't yet comment other posts (namely this) so I have to post the solution as a separate answer.

Golfed away 7 bytes by ovs

p x=reverse x++x!!0:x
u m|n<-m+div(m-1)7=[(' '<$[z..n])++p(drop(n-z)$take(n+1)$cycle" vibgyor")|z<-[0..n]]

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(Old version, 113 bytes)

Haskell, 114 110 101 bytes

Thanks to [nimi][1] for -4 13 bytes!

f n=""#(n+1+div n 7)
w#0=[]
w#n|x<-cycle"r vibgyo"!!n=((' '<$[2..n])++reverse w++x:x:x:w):(x:w)#(n-1)

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R, 130 bytes

function(n,k=n%/%7*8+1+n%%7,a=el(strsplit(' vibgyor'/k,'')))for(i in k:1)cat(d<-' '/(i-1),a[c(k:i,i,i:k)],d,sep='','
')
"/"=strrep

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• -6 bytes thanks to @JayCe

Java (JDK 10), 184 bytes

n->{int h=n+n/7,i=h+1,w=i*2+1,j,k=0;var o=new char[i][w];for(;i-->0;o[i][w/2]=o[i][w/2+1])for(j=w/2;j-->0;)o[i][j]=o[i][w+~j]=i<h?j<1?32:o[i+1][j-1]:" vibgyor".charAt(k++%8);return o;}

Try it online!

Prints an extra leading and trailing space for each multiple of 7.

Explanation

n->{                             // IntFunction
 int h=n+n/7,                    //  Declare that height = n + n/7
     i=h+1,                      //          that index  = h + 1
     w=i*2+1,                    //          that width  = (h+1)*2+1
     j,                          //          j
     k=0;                        //          that k      = 0
 var o=new char[i][w];           //  Declare a 2D char array
 for(;                           //  Loop
   i-->0;                        //    Until i is 0
   o[i][w/2]=o[i][w/2+1]         //    After each run, copy the middle letter.
 )
  for(j=w/2;                     //   Loop on j = w/2
   j-->0;                        //     Until j = 0
  )                              //
   o[i][j]                       //    copy letters to the left side,
    =o[i][w+~j]                  //      and the right side
    =i<h                         //      if it's not the last line
     ?j<1                        //        if it's the first (and last) character
      ?32                        //          set it to a space.
      :o[i+1][j-1]               //          else set it to the previous character on the next line.
     :" vibgyor".charAt(k++%8);  //      else assign the next letter.
 return o;                       //  return everything
}

Credits

• -2 bytes thanks to Kevin Cruijssen

Canvas, 29 28 26 bytes

７÷Ｕ＋｛ <ibgyor＠¹×／ｎ｝⇵Ｋ２＊∔─↶

Try it here!

Explanation:

7÷U+                          ceil(input/7) + input
    {             }         for n in 1..the above
      <ibgyor@                in the string " <ibgyor", pick the nth character
              ¹×              repeat n times
                /             create a diagonal of that
                 n            and overlap the top 2 stack items (the 1st time around this does nothing, leaving an item for the next iterations)
                   ⇵        reverse the result vertically
                    K       take off the last line (e.g. " <ibgyor <ib")
                     2*     repeat that vertically twice
                       ∔    and append that back to the diagonals
                        ─   palindromize vertically
                         ↶  and rotate 90° anti-clockwise. This rotates "<" to "v"

25 24 22 bytes after fixing that ｍold should cycle if the wanted length is bigger than the inputs length and fixing ／ for like the 10th time

05AB1E, 32 31 23 bytes

.•VvÈ©•¹∍¬„ v:Rηε¬ý}.c

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-1 thanks to Kevin Cruijssen and -8 thanks to Adnan

Explanation (Stack example w/ input of 3):

.•VvÈ©•                  # Push 'aibgyor'           | ['aibgyor']
       ¹∍                # Extend to input length.  | ['aib']
         ¬               # Push head.               | ['aib','a']
          „ v:           # Replace with ' v'.       | [' vib']
              R          # Reverse.                 | ['biv ']
               η         # Prefixes.                | ['b', 'bi', 'biv', 'biv ']
                ε   }    # For each....             | []
                 ¬ý     # Bifurcate, join by head. | ['b','b']       ->    ['bbb']
                                                    | ['bi','ib']     ->   ['biiib']
                                                    | ['biv','vib']   ->  ['bivvvib']
                                                    | ['biv ',' vib'] -> ['biv   vib']
                     .c # Center the result.        | Expected output.

Red, 153 bytes

func[n][r: take/last/part append/dup copy"""roygbiv "n l: 9 * n + 8 / 8
repeat i l[print rejoin[t: pad/left take/part copy r i l last t reverse copy t]]]

Try it online!

Slightly more readable:

f: func[ n ] [
    r: copy ""
    append/dup r "roygbiv " n
    r: take/last/part r l: 9 * n + 8 / 8
    repeat i l [
        print rejoin [ t: pad/left take/part copy r i l
                       last t 
                       reverse copy t ]
    ]
]

Jelly, 28 bytes

:7+‘“ vibgyor”ṁµṫJZz⁶U;"⁸ŒBṚ

A monadic link accepting an integer which yields a list of lists of characters.

Try it online! (footer joins with newline characters)

Or see the test-suite.

How?

:7+‘“ vibgyor”ṁµṫJZz⁶U;"⁸ŒBṚ - Link: integer
:7                           - integer divide by seven (number of full rainbows)
   ‘                         - increment (the input integer)
  +                          - add (gets the number bands)
    “ vibgyor”               - list of characters = " vibgyor"
              ṁ              - mould like the result above (as a range)
               µ             - start a new monadic chain
                 J           - range of length
                ṫ            - tail (vectorises) (gets the suffixes)
                  Z          - transpose
                   z⁶        - transpose with filler space character
                             -   (together these pad with spaces to the right)
                     U       - reverse each
                             -   (now we have the left side of the rainbow upside down)
                        ⁸    - chain's left argument, as right argument of...
                       "     -   zip with:
                      ;      -     concatenation
                             -   (adds the central character)
                         ŒB  - bounce (vectorises at depth 1)
                             -   (reflects each row like [1,2,3,4] -> [1,2,3,4,3,2,1])
                           Ṛ - reverse (turn the rainbow up the right way)

Jelly, 31 bytes

:7+‘µ“ vibgyor”ṁṚ,Ṛjṛ/ƲƤṭ"ḶṚ⁶ẋƲ

Try it online!

Check out a test suite!

ಠ_ಠ This is overly complicated because Jelly doesn't have a centralize function...

Charcoal, 30 bytes

↶≔… vibgyor⁺²÷×⁸⊖Ｎ⁷θθ⸿Ｅθ✂θκ‖Ｏ←

Try it online! Link is to verbose version of code. Explanation:

↶

Change the drawing direction to upwards.

≔… vibgyor⁺²÷×⁸⊖Ｎ⁷θ

Calculate the height of the rainbow and repeat the literal string to that length.

θ⸿

Print the central line of the rainbow.

Ｅθ✂θκ

Print the right half of the rainbow by taking successive slices and printing each on its own "line".

‖Ｏ←

Reflect to complete the rainbow.

Python 2, 84 82 81 bytes

-2 bytes thanks to ElPedro.

n=input();n+=n/7;w=''
while-~n:w+=" vibgyor"[n%8];print' '*n+w+w[-1]+w[::-1];n-=1

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JavaScript (ES6), 100 bytes

Returns an array of strings.

f=(n,a=[i='   '])=>++i<n+n/7?f(n,[c=' vibgyor'[i&7],...a].map(s=>c+s+c)):a.map(s=>' '.repeat(--i)+s)

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Python 2, 108 bytes

n=input()-1
n+=n/7+2
o=[]
for s in(' vibgyor'*n)[:n]:o=[s+l+s for l in[s]+o]
for l in o:print l.center(n-~n)

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Python 2, 132 131 bytes

def f(n):
 t=n+n/7;s=('vibgyor '*n)[:t];r=[s[~i:]+t*' 'for i in range(t)]
 for l in zip(*r+3*[' '+s]+r[::-1])[::-1]:print''.join(l)

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Saved:

• -1 byte, thanks to Jonathan Frech