Why is my right hand often swollen when I wake up? It's not numb/tingling and I've had no injuries.

Positioning. This is probably due to positioning, that is to say you maybe sleeping on that hand. Try to change your sleeping position and see if that helps.
Maybe vein's blocked. A person who goes to bed fine but wakes up with a swollen right hand but a normal left hand, on repeated occasions, has an interesting problem worthy of a doctor's evaluation. Maybe the veins draining blood from the r hand or arm are compressed or somehow blocked during sleep. Maybe the nerves to the r hand are malfunctioning and signaling the arteries to let more blood flow to the r hand.