I agree with
@ewoodrick in that you nearly certainly have a 200a capable service. Better pictures may be necessary to 100% verify and I actually would also probably take the panel cover off and verify the main feed wire gauge. You don't have a whole ton of 240v loads in this panel and it is a modern panel so I see no issue with a 14-50 receptacle and circuit for a UMC Gen 2 to draw 32 amps. An official load calculation would need to be done, but you could likely handle a 60a circuit (48a usable charging) with a Tesla Wall Connector which would let you charge the car at its max rate (I have a M3 LR myself with a 60a circuit into a Wall Connector). You may even be able to support more than a 60a circuit so that as others have mentioned, you could load share to a second Wall Connector for a second vehicle.
I would not say it is a forgone conclusion that you could not get two 14-50 receptacles out of this box. It all depends on the load calculation. Note that if you are just using a UMC Gen 2 which can only take 32a continuous then you can still install 50a circuits on a 14-50 but you only have to calculate it for the intended load. So it is well within the realm of possibility the load calcs would allow two 14-50's. @
Whosyourbaba also actually has sufficient space in this panel to install two new 240v circuits. It would just require moving one 120v circuit from one side of the panel to the other in order to get two double wide breaker positions available adjacent each other. This would fill the panel up completely. This panel won't be able to accept any tandems either since it is likely limited to 40 total positions of which 36 of them are full. It does not appear to accept tandems in any slots.
So the good news here is that you have options and you will be able to do EV charging off of it. The question though is what solution to go with out the gate. It kind of depends on how much future proofing you want to do and how cost sensitive you are (as well as how difficult it is to physically install where you want the charger or chargers). Oh, and we have not asked you how many miles a day you and your family members plan to drive once you get a Y as well.
Bottom line is I personally would nearly guaranteed be installing a Wall Connector since I have one and I absolutely love it. I love knowing that I can charge at the max rate when I need it, and the car can take a shocking amount of power when pre-warming the car and battery in cold weather (which is nice to do off shore power rather than the battery). Though parking it inside would make a huge difference (mine is outside). The question if you go with a Wall Connector is how large a circuit to install and how to physically wire it if you think there is a large chance of getting a second Tesla soonish.
I would likely do surface mount conduit (EMT) as NM cable (romex) gets more challenging for the larger circuits especially as it can only use the 60c rating per code which requires larger wire than if you were in conduit and could use the 75c rating.
One very viable option might be to just do a 60a circuit in 3/4 in EMT conduit, but run it through a junction box with a service loop of wire at a convenient spot to "T" off to a second Wall Connector (also would need to account for a way to run the low voltage control wire). Then if you added a second Wall Connector you could just have it share the same 60a circuit. Either car could have access to the max 48a charge rate by itself, or if both needed to charge they could both charge at 24a. It would be an exceedingly rare situation to have both cars come back home near empty on the same day and then need to get back up to 100% for the next days driving.
If you wanted to future proof further, you could use 1" conduit and optionally larger wire upfront which could allow you to share more than a 60a circuit in the future. If you ran smaller wire today (but still 1" conduit) you could easily pull it out later and pull in larger wire if that came to be. Naturally you would need to make sure whatever you do fits within the "load calculations". (no point in running larger conduit or wire if your main service feed won't support it without an expensive major upgrade)
Sorry for the wall of text. Good luck and please report back what you end up doing! Let us know if you have any further questions!
