Maybe something has changed. When Diane and I looked for companies to rent us an air brake equipped straight truck, the easy part was finding the companies. The challenge was finding a company that would rent us a truck when neither of us had a CDL, but only permits.
In Minnesota, truck driver training companies are allowed to advertise in the CDL manual that is distributed free by the state. Through those ads, we found a truck to rent for the road test.
You might try going to the testing center in your state and looking around first thing in the morning. People take CDL tests often. Rental trucks may show up with them.
Also check the bulletin boards there for ads. Truck driving schools may be an option. There may be some that will rent you a truck, but unless they are willing to deliver it to the test site, you cannot legally drive it on a permit alone. You need a licensed CDL holder in the truck with you. Again, unless something has changed.
With the truck we rented, the company brought it to the test site the morning we needed it. One of their instructors took us out for an hour each so we could get used to the truck before using it for the road test. Allowing students to drive the truck before taking the test helps keep the inspector's teeth marks out of the dash board.