DOS_deck offers free, all-timer DOS games in a browser, with controller support

Doom is just the shareware version, just like most of the others (some already called with that fancy modern name "demo"). Some are freeware, some have been released into public domain after they went out of sale.

Is it the full game or the shareware? In another post a few days ago they said it they were only offering shareware versions. Can't check now, I'm on mobile.

It wants a code for level selection. You get the code for level 2 once you finish level 1, and so on. So just start with level 1 (F1).