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The Peaceable Journey Law is the only regulation I know of.
TITLE 18--CRIMES AND CRIMINAL PROCEDURE

PART I--CRIMES

CHAPTER 44--FIREARMS

Sec. 926A. Interstate transportation of firearms

Notwithstanding any other provision of any law or any rule or
regulation of a State or any political subdivision thereof, any person
who is not otherwise prohibited by this chapter from transporting,
shipping, or receiving a firearm shall be entitled to transport a
firearm for any lawful purpose from any place where he may lawfully
possess and carry such firearm to any other place where he may lawfully
possess and carry such firearm if, during such transportation the
firearm is unloaded, and neither the firearm nor any ammunition being
transported is readily accessible or is directly accessible from the
passenger compartment of such transporting vehicle: Provided, That in
the case of a vehicle without a compartment separate from the driver's
compartment the firearm or ammunition shall be contained in a locked
container other than the glove compartment or console.

There is NO regulation in the FMCSR -Federal Motor Carrier Safety Regulations.
I carried in a big truck for decades. A law...no.
An interpretation 'of the law' by some LEO...YES.
If you are dispatched to go into Canada.....I suggest you better not have
a gun in your possession .....If you're dispatched to a military base, you could have some problems, although a camera and a laptop are just as bad. [I used to frequent military bases, carried, and locked everything in a secure cage at the checkpoint, picked it up on the way out]
Get caught with a gun in your truck in California.....we may never hear from you again.
California will not even allow a baseball bat in a commercial vehicle.....not for any reason.
Legal ?? I doubt it.....enforced...you bet.
New Jersey.....Chicago....hmmm....you could be in real trouble.....
Legal ? ENFORCED by their own interpretation !!!!
You are guilty....until you prove your innocence .....
I drove for 43yrs and owned a transportation company for
most of them. I've not only heard the rumors, and the stories,but have seen it.
And there are many others of us....right here on this forum that also have.
Just for reference, many years ago you had to acquire a permit from each state
to carry a weapon in your truck. Renew it each year. Connecticut was by far
the easiest to get....5$ in the mail.