Your AC load is 1/2 of your available supply!
No, this is not true. The AC needs a 50a breaker to be able to handle the startup surge current of the motor. The motor will have separate thermal protection which protects the wiring of the circuit.
You do not calculate the usage of that circuit as the full 50a of the breaker. Continuous run current is likely to be more like 20a or something (find the nameplate of your AC for details).
Almost every breaker is doubled up already, it's amazing you don't pop a breaker every day!
If you had your car charging in the summer off a 50 amp breaker (which there is technically room for), with A/C on and a few lights or an oven or a microwave, your main breaker would trip and your house would go dark.
I would upgrade to 200 amp service, even without the need for a charger (a few thousand $). If the panel is near the garage, you could even have them run some thick wire and get a 90-amp line in the garage in case you ever want a beefed up HPWC.
If you don't want to upgrade service, I'd stick to Level 1 charging off a 110v outlet.
While a 100a service is very small, there are extremely minimal loads in this panel. The heavy hitter is the AC and the dryer.
@bottomsup stated they will be using a gas dryer, which removes that from the equation (which is a great way to free up capacity for EV charging btw). I am not surprised at all that 100a has been sufficient.
A NEC load calculation needs to be run to figure out how much capacity might be available in the panel. One of the factors that goes into that is square footage. How big is the house?
That panel is a relatively modern one (though no spring chicken). I don't see safety concerns with it.
There is a very good chance you could actually support a NEMA 14-50 on a 40a or 50a circuit which would let you use a UMC Gen 2 at the 32a charge rate (note that load calculation wise, whether it is 40a or 50a does not matter since the intended load is a UMC Gen 2 at 32a - so I would always wire for 50a).
From the external sheet metal, it would appear that the top two breaker positions are open which would trivially allow you to install a new 240v breaker, though there could be no bus bar behind that. I would need to see the spec sheet (is there on on the inside door lid?) or a picture with the panel cover off to know.
If it were my house, I probably would want to upgrade beyond 100a though. It is annoying that it is embedded into the outside of the building. That might make swapping it slightly difficult. Can you provide more pictures (zoomed out of the panel outside, etc...). I am wondering what gauge the feed wires into the utility panel are and what size conduit they have? It might be really trivial to upgrade to a 200a service if the right size conduit is in place already (likely is).
If you were really lucky there might be a way to replace the "guts" of the panel to at least 125a or maybe 200 without replacing the entire panel? Unlikely though.
Even if load calculations won't allow a full 14-50 for charging at 32a, you likely could at least do 30a (for a 24a charge rate) if you decommission that dryer.
Good luck and please report back on what you end up doing!