Several cab companies have tried to have "order by phone" apps in the UK to compete with Uber, they all suck. Cheaply made, basically just website forms, because they're not being made by people with skills, knowledge, and long term investment in the maintainence and expansion of the app. They're being contracted out to be built once-and-done by people who don't understand what they're contracting, and it shows.
You could construct an Uber-style app that Cab Companies pay a fee to be listed in and integrate with, and then aggregates requests between Cab Companies, but that's arguably just an extra level of indirection so quickly raises the question "why not go straight to the drivers?".
(In the UK, Taxi Drivers are entitled to minimum wage etc as being employees of the cab company, which is why Uber demanding exclusivity from them but not actually treating them like employees landed them in court here.)
If you're renting a cab from them and it has their branding and you have obligations of how much you work to them and people call that company to arrange for you to pick them up and then they take a portion of that...that sounds like *employment* to me and you should have the things like minimum wage, sick pay, paid holiday time, protection from unfair dismissal etc that this should entitle you to (yes, USA doesn't always have all of those State-by-State but it should).
If they're just providing a vehicle rental service, and you have to do all of the rest of the work yourself, then that's being self-employed and you are paying them for a service as part of using that service in your business. The business world is held together by middlemen services nowadays.
The problems with things like Uber is they want the benefits of having employees (such as exclusivity, working hours for coverage etc) without the obligations (minimum wage etc).