I've been looking all over and I can't find an answer anywhere. If my teammate and I are taking our turn at the same time, and we both select the reinforcement card, who actually gets it? The first person to take it or the last?

Person who clicks submit moves first I think

You can't commit if you are using a card that a teammate, who entered his moves already, will use that turn.

So definitely first come, first serve. But on teamgames you should talk together anyways to find out who needs the card more