Could tendonitis turn into arthritis over time?

NO: Tendonitis is by definition an inflammation or injury of a tendon and arthritis is a disorder of the joing. So they involve two distinctly different structures and don't "turn into each other". Having said that, they can certainly co-exist and one can affect the other for sure.

Possibly but rare: Tendons support bone and joints. If the tendons do not function well they could cause joints not to function well. This could occur for eample in the ankle and knee and possibly lead to narrowing of the joint which could increase the risk of arthritis. If tendons cause significant pain, joint motion could be limited which obviously could affect joints but i still believe arthritis will be rare.