Technically, it's possible for the onboard software system to be designed to allow 3rd party apps to run, and prevent them from accessing critical car functions - like what's done on smartphones or with commercial-grade operating systems.
But, for that to work, it almost has to be designed into the software system from the beginning.
Tesla had originally planned to have a 3rd party app store - and Musk even made some comments (in Europe?) about a new operating system release that would support 3rd party apps - but there hasn't been anything new on this subject since then.
Even if there weren't many 3rd party apps, among the Tesla owner community, there is probably enough support to get a few apps developed.
But, if Tesla hasn't designed their software to support a "protected" environment for running 3rd party apps, we're unlikely to ever see it, unless Tesla decides to re-implement their onboard software (which other companies do periodically with their major software systems).
It seems more likely we'll see a "screen mirroring" solution - that keeps the apps running on another device, and the Tesla console is only used as a display and limited user interface - without any 3rd party access to the car's critical functions.
This is an area where Tesla is clearly behind the other manufacturers...