Should i be running with tendonitis?

Not without exam: If your tendonitis is in any of the lower extremity tendons you should be supervised and advised by a physical therapist or physician.

Need to be more: Specific, as in what type of tendonitis you are referring to. In general with most forms of lower extremity tendonitis(in fact, what's often diagnosed as tendonitis is more often tendinopathy - noninflammatory degenerative tendon changes due to chronic overuse) requires some form of activity modification/restriction, often that includes choosing alternatives to running or at least reduced mileage