Is it okay to play the piano without a bench? I don't have enough money for a bench, so I have to use my bed. I don't have very good posture as a result.

This is not an ideal situation, particularly if your posture is poor. Your bed likely isn't at the right height for you to play the piano properly. A bed also likely doesn't provide enough support. Look for a used bench at thrift stores, or talk to musicians you know – they might know where to find a used bench. Music stores also often have bulletin boards where customers advertise used music equipment and instruments for sale, so you might check there. You don't even have to get a piano bench specifically, but you should try to find a hard chair that's at the right height for you. Playing with poor posture will put you at higher risk for back injuries and other repetitive motion injuries.