a transcript only contains the last 4 didgits. don't know if a full copy of return would have complete SSN
I'm also guessing that you don't have access to the TT a/c that was used last year. that only leaves you the option of mailing it in with a cover note explaining that your wife will not provide you with her SSN. provide her full name and address.
this is where you can get burned if she already filed. you must both use the same method either itemized or the standard deduction. if she has already filed, hopefully she'll tell you what she used. otherwise you get to choose unlesss you both can agree on what to use.
if she filed first and used standard (or itemized) and you used itemized (or stndard) then the IRS will probably change your return. if you itemize , there should be some sort of agreement who gets what. the IRS doen't like it if both claim the same deductions but in some cases they never know.