Can I get an iPhone 5 without a contract?

I have an iPhone 4 and contract with AT&T but my phone broke and I can't get a new one unless I renew my contract. Can I buy an iPhone 5 and put the SIM card in it? (If so how do you do that on iPhones?)And where could I buy it?

You could. The price is around $600 or so for buying without a contract. There's no sim card in the iPhone. As to where to buy the phone, check the stores who carry AT&T since the phone will need to work on their network.