You want to use the VNA against the final system configuration... or at least as much as practical. Why? The antenna's environment and transmission line will impact its characteristics.
It's pretty trivial to place and retrieve an antenna on a magmount on the roof of the car. So in this case, you simply plug the VNA in to the antenna in place of the radio. If you find you need to adjust placement of the magmount or length of the antenna, just grab it down and make the change. It's more challenging to do the same with a tower mounted yagi, so compromises will be made in testing configurations.
I suspect what you're finding is the need to place an appropriate choke at the antenna feed point. Dropping frequency by adding transmission line may indicate that the outside of the shield is a meaningful component of the system, which it should not be. Adding a choke at the feed point will help address the situation if that is indeed the cause.
When it comes to calibration, you'll want to do so with everything in the system attached to the VNA, replacing the antenna with the open/short/50Ω slugs that came with your VNA.