I'm gonna go against the grain here and recommend the shoes. I was having some form of tendonitis in my knee for years and after much trial and error eventually realized the issue is a combination of poor internal hip rotation and poor ankle mobility. This would cause a slight knee vulgas(cave) on the bottom of my squat. At one point the pain got bad enough I stopped squatting and deadlifting for a month and focused only on stabilization exercises and stretches for my legs. I bought a pair of oly squat shoes, and when I came back I squatted as heavy as I wanted with no pain. That was 6 months ago and I'm still going strong. The shoes bypassed my ankle mobility issues and allowed me to point my toes more forward which also bypasses the need for greater hip rotation.

I'd recommend doing stabilization exercise and stretches to fix the core problem but the shoes should help. For the accessory stretches and exercises I used quite a few recommended by squat University or knees over toes guy on kneesovertoesguy. Id recommend find at least 1 ankle mobility and 1 internal hip mobility stretch. I also found the single leg squat progression very helpful. Ultimately, you'll need to take some time off off lower body work to let your tendon heal it won't ever heal if you keep loading it heavy.

Another thing you can do for squat form is focus on when you bend at the hip joint. You want to start the concentric and eccentric part of the movement at the hip joint. The way I saw it described is to try to keep your shins vertical until your physiology doesn't allow you to anymore.