FAFSA and a recent permanent resident
For the following, I will assume that you were on a F1 visa from arriving in the US until after your 26th birthday. In that case, you should have been exempt from selective service. If you had any kind of change in immigration status from a non-immigrant visa (such as having the F1 visa expire) during this time, well, that could complicate things and you really should talk to your university about it.

The Selective Service System's page for Men - 26 and older suggests that you can get a status information letter to prove that you're exempt. You could ask your university's financial aid office if it's required in your case or not.

Reading more on the Status Information Letter page, I found the following (which suggests it isn't required):


  Financial aid officers:
  If the man falls within one of the following categories, a status information letter from Selective Service is NOT required if he can provide a copy of supporting documentation proving his case to your satisfaction. &lt;...>
  
  Non-U.S. male on a valid non-immigrant visa
  
  For example, if the man entered the United States as an F-1 student visa and remained in that status until his 26th birthday, he would need to provide documentation indicating that he was admitted on an F-1 visa and attended school full-time as required.