Texas has no minimum age for possessing a handgun. The Motorist Protection Act applies whenever you are in or en route to a motor vehicle that you own or control.
So: yes, under 21 can lawfully possess a concealed handgun in a motor vehicle that they own or operate, so long as it is not in plain view, they are not prohibited by law from possessing a firearm, are not engaged in criminal activity other than a Class C traffic violation, and are not a member of a criminal gang. Texas law makes no mention of loaded or unloaded.
There is a federal law restricting possession of handguns by anyone under 18, but that doesn't apply to your question.