There website says 1 year experience. But like everyone else they need drivers. So if they are a company you would like to work for. Call them, put in a application. The worst thing they can say is no.
Yes and no. They would hire you as a dock worker and put you into the driver training program. The bad news is that you wouldn't be driving right away, but the good news is that you'd make a good wage in the meantime and before you know it you will be making far more than someone who was hired straight out of school by Swift or Werner, etc.
Unfortunately, there's pretty much no way around getting trained at a company by a driver trainer (whether it's a starter OTR company or an LTL company with a driver training program). You may think you know how to drive a truck, but all you really know fresh out of school is how to pass a CDL exam. I was extremely glad to have 12 years of experience before I started as a P&D driver with Dayton Freight, because I was able to calmly deal with incredibly stressful situations that required a high degree of skill. Anyone who thinks they can do that job straight out of school has no idea how difficult it would be. You'd be set up to fail.
If I could do my career all over again, I would have skipped all of the bad jobs and gone the dock-to-driver route. Unfortunately, I also know how that new CDL beckons you to get out there and use it right away. Good luck, whichever route you take!