Can one work around shoulder tendonitis in the gym?

Yes: Shoulder tendinitis is a very common problem and can be treated with ice, nsaid's and activity modification. As far as exercising at the gym is concerned, try to avoid overhead lifting. Military/shoulder press and push-ups should be avoided. Add internal/external rotation exercises to your routine and always ice after exercise. After 4-6 weeks you should feel better, if not see an orthopod.

Avoid shoulder : I would do legs and abs and avoid using the arm to do weights for now and only do the shoulder exercises your therapist has taught you until the pain is gone.