Yes, the following are some of the ways that you may be able to start a moving business without owning your own moving truck.
Pickup Truck Rental: You can rent a pickup truck from locations such as U-Haul for smaller moves.
Truck Rental: Moving trucks can be rented for short periods of time. Moving trucks may offer moving vouchers to inexperienced movers, which provide discounted rates on the first several jobs completed for new movers.
Van Rentals: Moving vans can also be rented for short periods of time if customers do not want to purchase one.
Act as Freelance Movers: Moving companies will sometimes hire freelance movers.
Partner with Moving Companies: Contact local moving companies and inquire about becoming their partner, or hiring them for some jobs at least some of the time.