You don't need an actual copy of the 1099-R as you don't need to attach it to your amended 1040X tax return. So I don't see why you couldn't amend the 2022 tax return now as you should know the employer name, address, ID#, amount, distribution code, etc on the 1099-R. So just enter them into TurboTax. Based on what you wrote in the OP, I'm pretty sure your wife's 2023 Form 1099-R will have a P Distribution Code in Box 7. But you should contact HR at her former employer to make sure that's the case.
I'm no TurboTax expert. Here's a previous answer to a similar question:
https://ttlc.intuit.com/community/after-you-file/discussion/where-do-i-enter-1099-r-code-p-when-amen...
A 2018 1099-R with a code "P" in box 7 is taxable in 2017, the year the contribution was made, not 2018.
You must amend 2017 to report it. The 2017 1099-R interview will say that code P means "Return of contribution taxable in 2016", but the interview will ask if this it a 2017 1099-R or a 2018 1099-R. Say 2018 and the "taxable in 2016" becomes "taxable in 2017" (the year advances by 1 each year).
It is not reported in 2018 unless there were earnings in which case the earnings should be reported on a separate 2018 1099-R with a code 8 on box 7 to be reported in the year the earnings were returned.
The end result is that you will increase taxable wages by the amount of the excess 401(k) DEFERRAL in the amended 2022 tax return on Line 1 of the 1040. The earnings are taxed in 2023.
https://www.irs.gov/instructions/i1099r#en_US_2023_publink[phone number removed]
Excess deferrals.
Excess deferrals under section 402(g) can occur in section 401(k) plans, section 403(b) plans, or SARSEPs. If distributed by April 15 of the year following the year of deferral, the excess is taxable to the participant in the year of deferral (other than designated Roth contributions), but the earnings are taxable in the year distributed.
