Right hand and shoulder get stiff have difficulty moving joints after long stretches at computer. what could cause this?

Typically ergonomics: Most likely an ergonomic problem. You need more frequent breaks, changes in position, and hands-free solutions if possible. Also consider keyboard and mouse changes. If the pain does not disappear, the cause needs to be identified. It can be tendon/joint problems (arthritis, tendinitis, etc.) or nerve disease (carpal tunnel syndrome, etc.). Many of these conditions are treatable