Yes you will need to amend 2020 return, the unpaid portion of the first-time home buyer credit was due in the year the home was sold.   Ask the client if by chance IRS caught this problem and sent a notice.

https://www.irs.gov/taxtopics/tc611

Good thing your client has switched to you as their preparer, last one missed something important.

As a side note, I never prepare a new clients return without a copy of the prior year return.