Contests or sweepstakes usually have language that says "open to U.S. residents" and legally you qualify because you live in the U.S and have a valid ID. I'd say just give them your ID and EAD and see what happens. They may have asked for "residency card" because I assume they are buying your airplane tickets and will need another form of government ID.
Is it anything like the contest here
https://static01.cirquedusoleil.com/...962.1549397603