As I never left my tack at the stables, but kept it at home, I included it in my household insurance, but things may have changed.
I would ask for quotes from household and yard insurance, but I am pretty sure it wouldn't be covered in your car.
A lot depends on the security at the place you keep your tack, you are the best person to judge the most secure place.
As my tack included a sidesaddle, it caused a lot of interest and other members of the yard asked to try it, but I would only allow them to do so under my supervision, especially as at the time the saddle was worth more than the horse I was riding!
Even now, I would take a sidesaddle home, especially as they cost in the region of £3,000 for a new one!
Cheers, Midori