Octave, 88 72 bytes

72 bytes, Saved 16 bytes thanks to @ceilingcat

Try it online!

function r=f(y)r=mod(fix(y/100)-fix(y/400)-fix(y/4)-y-3,7)*(1+i)+8+i;end

Original answer:

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function r=f(y)r=mod(floor(y/100)-floor(y/400)-floor(y/4)-y-3,7)*(1+1i)+8+1i;endfunction

Bash + GNU date + GNU dc, 27

Refactored from @Neil's answer to use dc to do the arithmetic.

date +8r%u-d7+f -d$1/3/8|dc

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Bash + date, 45 42 bytes

n=$[8-`date +%u -d $1/3/8`];echo $[7+n] $n

Try it online! Link includes test cases. Explanation: +%u gives the day of the week as a number from 1 for Monday to 7 for Sunday, which when subtracted from 8 gives the date of the first Sunday in November; 7 is then added to give the date of the second Sunday in March.

Sadly GNU date's weekday support only seems to work for dates relative to today.

Edit: Saved 3 bytes thanks to @DigitalTrauma.

Python 3.8, 46 bytes

lambda y:[n:=(y//100-y//400-y//4-y-3)%7+8,n-7]

An unnamed function that accepts the year and returns a list of the two days of March and November.

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Python 2, 44 bytes

Backport courtesy of Lucenaposition.

lambda y:(y/100-y/400-y/4-y-3)%7*(1+1j)+8+1j

An unnamed function that accepts the year and returns a complex number with the day of March as the real part and the day of November as the imaginary part.

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05AB1E, 16 bytes

тx·4)÷`I3Æ7%81S+

Port of @JonathanAllan's Jelly answer, so make sure to upvote that answer as well!

Try it online or verify all test cases.

Explanation:

т                # Push 100
 x               # Double it (without popping)
  ·              # Pop and double it again
   4             # Push 4
    )            # Wrap all three into a list: [100,400,4]
     ÷           # Integer-divide the (implicit) input-year by these three values
      `          # Pop and push all three to the stack
       I         # Push the input-year
        3        # Push 3
         Æ       # Reduce the stack by subtracting: y//100-y//400-y//4-y-3
          7%     # Modulo-7
            81S  # Push pair [8,1]
               + # Add both to the value
                 # (after which the pair is output implicitly as result)

Shell and zdump on FreeBSD version ≥ 13.2, 45 bytes

How is this still a thing? On a FreeBSD system you do not need to re‑invent the wheel:

zdump -ic`read y;echo $y,$((y+1))` US/Eastern

prints for input 2024

TZ="US/Eastern"
-   -   -05 EST
2024-03-10  03  -04 EDT 1
2024-11-03  01  -05 EST

Spaces are horizontal tabs so their apparent widths depend on your tabs setting. If this concise format is too cryptic for you, try the ‑Verbose format:

zdump -Vc`read y;echo $y,$((y+1))` US/Eastern

prints for input 2024

US/Eastern  Sun Mar 10 06:59:59 2024 UT = Sun Mar 10 01:59:59 2024 EST isdst=0 gmtoff=-18000
US/Eastern  Sun Mar 10 07:00:00 2024 UT = Sun Mar 10 03:00:00 2024 EDT isdst=1 gmtoff=-14400
US/Eastern  Sun Nov  3 05:59:59 2024 UT = Sun Nov  3 01:59:59 2024 EDT isdst=1 gmtoff=-14400
US/Eastern  Sun Nov  3 06:00:00 2024 UT = Sun Nov  3 01:00:00 2024 EST isdst=0 gmtoff=-18000

The date format may differ depending on your locale.
_{NB: This “program” does (correctly) not work according to specification for years before DST was introduced (y ≤ 1917).}

APL (Dyalog APL), 17 bytes^SBCS

Full program. Prompts for year and prints (day of March, day of November).

8 1-¯7|1⎕DT⊂⎕11 1

⎕… prompt for number, and

…11 1 insert as first first element of list [y,11,1]

1⎕DT convert to day count since 1899-12-31

¯7| mod −7 of that

8 1- subtract that from 8 and 1

Attempt This Online! (embedded in lambda for ease of calling)

Jelly,  21 20  19 bytes

:4:“ıd¢‘_/__3%7+8,1

A monadic Link that accepts the year as a positive integer and yields the days of March and November as a pair of integers.

Try it online! Or see the test-suite.

How?

:4:“ıd¢‘_/__3%7+8,1 - Link: integer Year
:4                  - {Year} integer divide by four
  :                 - integer divide by:
   “ıd¢‘            -   code-page indices = [25, 100, 1]
                        -> [Year//100, Year//400, Year//4]
        _/          - reduce by subtraction
          _         - subtract {Year}
           _3       - subtract three
             %7     - modulo seven
                      -> Delta = (Year//100 - Year//400 - Year//4 - Year - 3) % 7
               +8,1 - add [8, 1]
                      -> [8+Delta, 1+Delta]

JavaScript (Node.js), 67 bytes

y=>[(g=d=>(s=new Date(y+99,-1186,d)).getDay()?g(d-1):s)(14),g(252)]

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Silly but works

JavaScript (Node.js), 60 bytes

not support some ancient years(1-2 digits)

y=>[(g=d=>(s=new Date(y,2,d)).getDay()?g(d-1):s)(14),g(252)]

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Charcoal, 30 bytes

ＮθＩ⁺⟦⁸¦¹⟧﹪⁻⁺⁴÷θ¹⁰⁰⁺⁺÷θ⁴⁰⁰÷θ⁴θ⁷

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

Ｎθ                              Input year as a number
            ⁴                   Literal integer `4`
           ⁺                    Plus
              θ                 Input year
             ÷                  Integer divided by
               ¹⁰⁰              Literal integer `100`
          ⁻                     Subtract
                     θ          Input year
                    ÷           Integer divided by
                      ⁴⁰⁰       Literal integer `400`
                   ⁺            Plus
                          θ     Input year
                         ÷      Integer divided by
                           ⁴    Literal integer `4`
                  ⁺             Plus
                            θ   Input year
         ﹪                      Modulo
                             ⁷  Literal integer `7`
   ⁺                            Vectorised add
     ⁸                          Literal integer `8`
       ¹                        Literal integer `1`
    ⟦   ⟧                       Make into list
  Ｉ                             Cast to string
                                Implicitly print

Python 3.8 (pre-release), 63 bytes

lambda x:(w:=7-date(x,3,1).weekday(),w+7)
from datetime import*

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The output is a tuple of the dates (the November date then the March date). This does not work for years >9999.

Perl 5 -MDate::Manip -a, 55 bytes

say ParseDate"$_@F"for"2nd sun in mar","1st sun in nov"

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Factor, 59 bytes

[ <year> march 2 over november 1 [ sunday-of-month ] 2bi@ ]

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[         ! a function
<year>    ! create timestamp from integer year input
march     ! change from jan 1 to march 1
2         ! push 2 to indicate we want second sunday
over      ! create a copy of the timestamp on top of stack
november  ! change from march 1 to november 1
1         ! push 1 to indicate we want first sunday

[ sunday-of-month ] 2bi@  ! change both timestamps to nth sunday of month

]         ! end function