Is it permissible to take out a personal loan to buy a taxi for the purpose of earning a livelihood and supporting the family, given the unavailability of an installment option?

A loan with interest is a prohibited usurious loan that one may not enter into except out of a recognized necessity. If you cannot find someone to sell you a car in installments, you can purchase another commodity in installments, then sell it and buy the car with its price. This is called Tawarruq, and it is permissible when there is a need for money.