Is it possible to get arthritis in your wrists?

Yes. Arthritis can affect people in the wrists with the aging process. There is also a higher likelihood of developing arthritis following a traumatic injury to the wrist (e.g. Fractures or sprains). Finally, some patients develop forms of inflammatory arthritis which can affect multiple articulations within the wrist.
Yes, with injury. Yes, many things can cause wrist arthritis. Most common is a ligament injury to the scapholunate ligament which causes instability of the scaphoid bone gradually leading to arthritis. Also fractures that disrupt joint surfaces can lead to arthritis. Less common causes are inflammatory disease such as rheumatoid arthritis, gout, and calcium pyrophosphate deposition disease *cpps).