Each player card on my Madden Ultimate Team has a limited number of games they can play on contract. I can get contract cards to increase the contract number, but is that card only usable one time on one player?
Yes, contract cards are single-use: one time only on one player only.
As with many other aspects of Ultimate Team play, there is precious little documentation available, but plenty of anecdotal evidence to sift through. In general, when you have a question about Ultimate Team, it's safe to assume that the least favorable aspect applies (having a player appear for one play will subtract one game from his contract, losing your connection during a game will count as if you had finished the game except that you won't get any coins).