Is there citizenship required or something that would prohibit an immigrant from becoming a teacher

No, there are no restriction like that. In fact, many schools like to hire teachers from other countries, so that their students are exposed to people from other countries and cultures. Of course you must have the proper visa. Also, if you're looking to teach in public school you'll need your teacher's certificate (private schools generally have no such requirement). See: Teaching Certificate

Nope, no rule requiring citizenship. Like ebaines says you do ned a teaching certificate for public schools. What level and subject do you want to teach and what's your background and training