As each location is different, it is best to contact your pick up depot beforehand to find out their specific policies.
Suppliers will generally allow small pets to travel along with you, though the car must be returned in the same condition in which it was rented out. Cleaning fees will apply. If there is any damage from the pet, you will be charged for the damage or charged the extra cleaning fee.
You are not allowed to travel with pets in a Thrifty vehicle.