Yes, Tesla will program keys, but you first have to create a Tesla account (if you don't already have one) and transfer the car into it by submitting a picture of the Title or valid registration, and a copy of your DL/ID, and the name on it MUST match the name on the title/reg. Then they will enable the app on the car and it can be driven. Then it's a simple matter to drive the car to Tesla and they will program a key.
I can also do this, but it's a bit more complicated, as it requires root access. Best bet is to go to Tesla.