The parallel is the search so many willfully submit to, though it is a greater invasion than a car search-- urine based drug testing for employment. The question, "what gives them the right" is answered by you want the job, they want to know that you are clean.
In this instance, the "larger company" could be treading on thin ice if they chose to search a car belonging the public (you did say the lot is open to the public) and the owner declined. They (the larger company) have no leverage on others except to tell them they should park elsewhere next time. To the contrary though, they can tell you to do what they want or get fired.
There have been companies who fired folks for off site conduct such as smoking at home.
We need much stricter employee rights in this country, and yes corporate/employee abuse needs to be curtailed. But, that isn't where one half or more of the electorate is as they don't give a dang about anyone's rights so long as their ox isn't gored.