Rules are probably different where you live vs. San Diego where I live, but here is the process as I understand it....First off, did you get a single charger in the car or dual chargers? What maximum charging rate are you shooting for if the panel can support it? What is the size of the main breaker in your existing main panel?
Here's my own thought process...
1) Before I pull a permit for the install, I need to perform standard load calculations for the panel that include the new charging station. A contractor can do this as well. Here in San Diego, they also allow actual values instead of estimated values, which may also give more headroom in the calculation.
2) You may want to think of these calculations as arriving at the "Sweet Spot" for the size of breaker you can install without major upgrades. For example, my panel may only support a 40a breaker due to the pre-existing load (without a major upgrade). I probably wouldn't want to do a major upgrade, so 40a may be the size I'm stuck with (or whatever the size is).
3) Once you have that maximum "Sweet spot" number that you can add, then you or the contractor can pull a permit and start the work. In your case, it looks like you're referring to the 20amp garage circuit pictured? That would work to charge the car, but the rate it would support is only 16 amps (or about 3.3 kW). You could probably install something larger since your house isn't that old. That's what the calcs will tell you....
4) Once you have the permit, then you or your contractor can start the work. It could involve upgrading the wire and breaker that you already have (Assuming your panel can support it).
So I think you have a few questions to answer first, and then go down the path to figuring out how big a breaker your panel can support...