Now I'm two months behind, and they are sending me letters threatening repossession. How can I prevent my car from being repossessed?
If you can, send your lender at least a partial payment to show your good faith on your loan and to reduce the chances of the situation escalating further.
Call your lender or go there in person immediately to discuss your situation and call every day until the situation is resolved. Document every conversation, including the name and title of each person you talk to, and send any letters certified mail, so you have proof of your efforts.
Be polite but firm about your need to somehow restructure your car loan and ask for the next person up the management chain until you get someone in a position that can help you.
Lenders would prefer to have their customers pay on their car loans rather than repossess their cars, and since your leave of absence is temporary, they should be more likely to work with you. Be prepared to provide documentation from your doctor regarding your length of absence and assurance from your employer that you do have a job to return to.
Rather than ask for your payments to be suspended entirely, it may be more effective to ask the lender for a reduced payment during your leave of absence, with the remaining amount spread out over the rest of your car loan.