What is the ruling on buying a car in installments from an institution that finances its operations by borrowing money from a bank with interest (riba)?

It is permissible to engage in transactions, both buying and selling, with a usurer. The Prophet, peace and blessings be upon him, used to deal with the Jews, even though they practiced usury. Borrowing with interest is a major sin, but that does not mean it is forbidden to transact with the borrower in buying and selling. Therefore, there is no harm for you in buying a car from this institution.