How do I handle not being able to talk to my boyfriend when he's not feeling "chatty? "

Not feeling chatty? Simple, ask him when does he has time to chat?
With some caution. I noticed you didn't ask; " how can I get him to talk when he's not feeling chatty". That's a good start. You can't make folks talk when they don't want to, you can only be available so when they want to talk you are there. As much as possible try to find out what drives him to not talk. He may or may not know. You could ask him about it when he is feeling "chatty".