Yes, that would work. You'll just want to make sure the adapter is rated for continuous use.
You may also want to open up that outlet (or have an electrician) and see if there's an unused neutral wire. If so, it would be easy to swap the outlet for a NEMA 14-50 and not have to use an adapter. Technically, you could do that even without a neutral wire since the EVSE doesn't use it, but it would be against code to have a NEMA 14-50 outlet without a neutral.
Basically - yes. I use a 6-50R receptacle at my in-laws place with an inexpensive 6-50P to 14-50R adapter and the included charger. The Ford Mobile Charger does not need the neutral wire, and very few other EVSEs need it either (and some can be purchased with a 6-50P plug right on them from the factory).
The only other thing I'd recommend confirming is to open up the electrical panel door and make sure that the breaker connected to that outlet is a 50A breaker - it is a not uncommon mistake/corner-cutting to install a 30A or 40A breaker instead. That looks relatively new so unlikely that they got that wrong (and passed inspection), but worth a glance.