"We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America."
Keeping its citizens safe would likely fall under general welfare.
Have you actually read Article I, Section 8? You'll have to squint very, very hard to see any authority to pass laws related to safety in it. That's because it's simply not there. The state governments, of course, do have that power, and the Congress arguably has power to regulate cars used in interstate commerce.
But there's simply no constitutional scope for safety regulation of cars built & used in only one state.
Instead of nitpicking, let's focus on the actual issue. The NHTSA didn't order Hotz to shut it down, it merely asked about the safety of the product, which led to Hotz shutting it down himself.
It also raises the question of what you think the Bill of Right's purpose is.
Which article of the Constitution is that? I don't seem to recall it.