Learn Hardcore Polish: Zwycięzca chce rewanżu, ale przeciwnik odpada z gry. - The winner wants a rematch, but the opponent is eliminated from the game.

So even though the ending may look feminine to an English-speaking learner, in this sentence it means the winner and is treated grammatically as a masculine personal noun. Its nominative singular form is zwycięzca.

For a specifically female winner, Polish often uses zwyciężczyni.

What form is chce, and who is doing the action?

Chce is the 3rd person singular form of chcieć = to want.

So:

ja chcę = I want
ty chcesz = you want
on / ona / ono chce = he / she / it wants

Here the subject is zwycięzca, so zwycięzca chce means the winner wants.

Why is it rewanżu and not rewanż?