USDOT for interstate commerce
There is no USDOT law or regulation that prevents you from carrying in or outside of your own rig while traveling interstate for commercial purposes. You must, however, comply with the laws of each state you travel to or through. Depending on what your home-state is, and whether or not you have a permit from that state (or whether or not one is required or offered for that matter), reciprocity agreements will come into play that you must educate yourself about the details of. If you're going to or through a state that doesn't have reciprocity with your home-state, you must follow the laws for transporting firearms rather than the carry laws. Transport laws often require locking your weapon(s) and ammo in separate compartments, and then some only require that your weapon(s) be unloaded without regard to being locked up or separated from ammo.
In short, you have a lot of homework to do. I suggest doing it before starting to carry in your rig. Some states may treat your sleeper during reset time like they treat hotel rooms, which is generally like they treat a person's home, but you'd have to revert back to relying on reciprocity agreements or transport laws, whichever apply to your specific circumstances, when you got back on the road in those states. And then some states won't treat your sleeper the same as a hotel room, so again, you have a lot of homework to do.
When I first started OTR I found that my best resource for these kinds of issues were trucker forums, not gun forums. DOT laws and regs will be cited chapter and verse on trucker forums, reducing your need to do all that research yourself. I know enough to know that DOT doesn't prohibit carry, but I don't recall the statutes and reg sections that prove that. Plus, carry laws from state to state have been changing a lot over the last several years, during which time I haven't been driving, so anything I even tried to say about individual state laws would be questionable for accuracy at best.
Good luck.
Blues
ETA: DOT doesn't involve itself in what policies your employer/contractor has in place. The only time DOT would report something firearms-related to your employer/contractor would be if they caught you with either an illegal gun or carrying illegally, the latter of which is unlikely that they would initiate the investigation as to legality unless it was a state like NY or NJ where nearly every out-of-state resident who's carrying, is carrying illegally.