Tennis elbow and carpal tunnel syndrome are frequent injuries in sports and work activities in general, and can are particularly debilitating for shooters. This is due to the repetitious nature of the activities coupled with forceful gripping. For instance, using the isosceles grip instead of of the Weaver can cause trouble in the elbows, because of gripping combined with elbow and wrist extension.
As an avid shooter and a health care provider who specializes in upper extremity rehabilitation I run across these issues quite a bit. (As well as finger amputations, sprains, burns, fractures, etc.) In fact, I have been toying with an article or brief training lecture about shooters' injuries: how to avoid them, how to compensate for them while they heal (and still be able to shoot) and how to correct them. It's just that every time this topic comes up people start to snore.
You can train hard, but you must stretch and rest appropriately in between sessions. Shooting a lot and then lifting weights to help with that shooting can easily be overdone. It's not all about strength, as endurance and flexibility are also key elements to shooting capability.
Grip strengthening exercises, and stretching and strengthening of the wrist and forearm flexor and extensor muscles are critical. (These are the so-called extrinsic muscles of the hand). Stretching and strengthening of the muscles of the rotator cuff and trapezius muscle in the shoulder are particularly important for shooters over 40.
It's all way too much to go into in a thread like this (pictures needed, etc.), but I'd be happy to help a shooter with injuries if they want to contact me via a PM or email.