Necroing this, as almost 2 years later I have encountered the exact same problem, and I have a vain hope that someone has maybe worked it out in that time.
Installed Destiny on the deck via Lutris, added the launcher as a non-steam game and set compatibility to use Proton, launches without issues.
Exact same phenomenon as BrokenBelle, I could enter my username/password to log into the game just fine with the virtual keyboard, then on a ship I am able to enter text into the password field, but not the party name field, rendering it impossible to start.
I've tried running the game from Desktop mode, making sure the Steam overlay is off, the last three Proton versions... I also tried mapping the "A" key to one of the back paddles, but that won't register anything in the Party Name field either.
Intriguingly, the Party Name field does react to the Escape key, so I can back out of entering text there, but that's the only key it seems to respond to. I've never seen such a bizarre problem.
Update: Also tried GEProton9-10 (the latest) in case that worked better than Valve's Proton version - no change.