Is it permissible for the indigent debtor, who owns nothing but a car whose value equals the debt, to sell it to settle the debt, or is he entitled to seek assistance from the Zakat institution?

You are permitted to request zakat if you are unable to pay your debt, and if the car is among your basic necessities, you are not required to sell it to settle the debt. However, if it is surplus to your needs and selling it would suffice to pay off the debt, then you must sell it, and you are not permitted to take zakat. If selling it would cover part of the debt but some remains, then you are allowed to take zakat for the remaining amount of the debt.