What does "publically accessible" mean? It doesn't mean "public", since the lot in question is private property. I guess you mean that it's connected to a road, so anybody driving by might be able to enter. If that cancels the owner's right to say who parks and who doesn't, then the same applies to your "publically accessible" driveway.
If they have the parking lot open to the public...
You use the word "public" as if it's magical. It isn't. And the parking lot isn't "open to the public"; the owner allows his employees to park their, providing they agree to abide by certain rules that he spells out in advance.
To make this clear, consider that he most certainly can fence in the lot and place a guard booth at the entrance. I'm sure you wouldn't deny that.
Now consider. Can the guard turn you away if you don't have an employee sticker on your windshield? Of course he can. Can he demand ID and check an employee list for your name? Yes, if the lot's owner instructs him to. And likewise, he can say, "Please raise your right hand and solemnly swear you have no weapons in your vehicle." You can refuse, of course. And he can refuse to lift the gate. You'll just have to park somewhere else.
The only difference is that your boss probably doesn't have a fence and a guard. He believes that you can be trusted to obey the lot rules without a guard watching you, and he's willing to use the honor system. Little does he realize, but you apparently believe that his parking lot isn't really his, but is rather "public property". Meaning, more or less, that it's really yours.
Logical... Controlled access to the lot should be required if they want to prohibit firearms on the premises in a motor vehicle.
I have noted many large commercial enterprises with expansive parking facilities have security that is virtually nonexistent. This conflicts with the right to defend yourself.
“A covenant not to defend myself from force, by force, is always void.” - Thomas Hobbes.“The right men have by nature to protect themselves, when none else can protect them, can by no covenant be relinquished.” - Thomas Hobbes.
It is sort of like the U.N. trying to force the Israelis not to defend themselves against the Phillistines when the U.N. won't do a damn thing to clamp down on them either...