Arguably Chelsea last month, but that'd make it a trick question in my eyes.
I imagine quite a few teams won't have played in the FA Cup after winning it, but I assume you really meant clubs rather than the 11 particular players, because that would make it a trick question as well really.
Is it a club that's changed it's name?
Any FA Cup other than THE FA Cup would make it a trick question. Beats me...
The only other thing I can think of is, as tell-me-more suggests, a club that changed its name. Could it be that The Wednesday won in 1896 and became Sheffield Wednesday the following season?