In a publicly accessible parking lot? No. Not without a warrent they cannot. I'd hold a different opinion if they were gating it and had armed security.
Private citizens have no duty to help you exercise your rights. Your ability to defend yourself going to and from work is not your employer's concern.
Nor can private citizen arbitrarily strip another of their Rights. No help is required.
You have every right to keep your gun in your car. However, you have no right to use your employer's parking lot in violation of his rules.
If they have the parking lot open to the public, then they also have to respect the equal Rights of those individuals that will come along with it. Just because my car is on their property does not give them de facto ownership of said car.
All they need is a sign saying "All vehicles subject to search." They probably wouldn't even need to do that- they could just post a security guard at the entrance and forbid entry to anyone who refused to consent to a search. Rules in employee handbooks authorizing searches of vehicles are common and fully enforceable.
Nor can private citizen arbitrarily strip another of their Rights.
You have no "right" to enter another's property in violation of their rules.
If they have the parking lot open to the public, then they also have to respect the equal Rights of those individuals that will come along with it.
So, do you have the right to hold a political rally in your employer's parking lot?
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.