If you are paying rent to occupy space in an apartment, you are indeed entitled to a key to the building and the apartment. Anytime money changes hands in exchange for an apartment or room, that is considered a business transaction. It's a contract. You may not be entitled to a mailbox key, but otherwise, how are you expected to enter your room if you do not have complete access to it?
Go online and find the web site for the building code in your state. Research the law. Then, you need to talk to your landlord as to why he or she is reluctant to give you a key. If he or she doesn't want to give keys out to someone who is living in the apartment, the landlord should reconsider renting out a room. And you should find someplace else to live.
Remember, after you find another place to live, you should give adequate notice to your landlord.