Is it possible an ankle boot can heal torn or pulled tendons or ligaments in an ankle? it's been 4 months and it hasn't healed because of movement

Yes: it is possible but should be in addition to Physicall therapy

It should help, but.: It would help to better understand the structures involved, which might require MRI. Once this is assessed, then the effectiveness of immobilization +/- therapy can be gauged. Depending if there is a tear, surgical intervention might be a better option, especially if the patient is in sports.