Can you lawfully carry a firearm on your person in a commercial vehicle?

Yes, in accordance with the laws for transporting firearms in each state which you're driving through. Your firearm is also subject to the laws of states you're stopping in, originating from, and destined for. E.g., if you're originating from, stopping in, or delivering to a state with a 10 round magazine limit, you're subject to that law.
It's a common rumor that a federal law exists which prohibits firearms in commercial vehicles - this is 100% false. There is nothing in the FMCSR or CVSS which addresses the matter, except to say that a loaded firearm may not be transported as part of the cargo.