Look for a expedited company that runs B trucks. Go to a truckstop and look around. There are companies like Panther and others that run those trucks. Also out of personal experience I worked for a company that did Audio/Visual for conventions...there are a lot more companies than you may first think that run a B. At the risk of people throwing stuff at me, you can always go to a moving and storage company. Start with your B, then go get your A learners or whatever in your state. Then use their equip and go test directly with your states dmv. Get your A and go elsewhere. Just options for you. Good luck.
With a class B , try to get into a construction outfit and run tri-axle , then maybe if they pull any trailers at all you could get your class A that way . All while making some coin.....
I don't many construction companies that won't hire with license infractions , hell if they did they probably wouldn't have any employees. Lol