Can you ride a bike with tennis elbow/lateral epicondylitis/tendonitis?

Not a doctor here, but I did leave an impression on an orthopedic specialist. 

The second shortest answer is YES with the absolute shortest being NO.  

Getting into longer answers...

What caused your tendonitis?  
Rest from what has caused tendonitis is recommended.  Increased blood flow while not continuing the cause might speed healing (plus, the anxiety of not working out is relieved).  

Riding aero or on the hoods?  As long as the ~90° elbow bend doesn't aggravate your condition, riding in aero on a trainer shouldn't cause much problem.  In this aero position, most of your weight is transferred through bone.  Riding on the hoods can be iffy.  If your elbow is flexed, you're putting *some* load on the tendon.  If your elbow is hyperextended, you're putting stress on parts of the joint that probably aren't happy already.  

Riding on the road?  Consider an analogy:  Would you drive yourself to the hospital/doctor with a badly broken right foot?  You can argue that as long as you use it gently, you'll be OK.  HOWEVER...the instant you have to jam on the brakes using that bum foot, you'll regret trying.  Same with riding your bike.   How will you react to sudden situations using your very angry elbow?  Both balance and reaction time may be off, and what under normal conditions would be no problem may now become a crash.  (I got lucky...continuing after my IMCdA crash wasn't the smartest decision even though it worked out OK.)