My experience was that it just doesn't like the Opera browser (and I couldn't login on Opera at all, always getting the 413 Bad Headers error or so). So if you can, try another browser. Worked a bit better for me on Chrome (where I conversely never logged in to the NHL.com account, dunno if that's actually required or not, maybe it has some internal cookies/IP check or something...). So I registered for an account on Opera but never truly logged in anywhere.
I don't remember if I submitted with Opera or Chrome though, went back and forth a bit... I feel like Chrome didn't even require login and it let me submit perhaps. On Opera it allowed me to submit on the second try (hence my name there lol)... Telling sign was if it told the "you must have at least x and y done before submit" at the bottom. If it was missing then the Submit button would not materialize properly no matter what I did. Also I couldn't make the initial picks, I had to randomize, and then I was able to switch the teams if I clicked around a bit.
The "best" feature on Opera is that when you scroll down the list of participants in the league, it just hard switches to the Invite share thingie with no way to scroll back up. Have to refresh and not scroll (the join league button is somewhere up there luckily if you don't scroll down).
So yeah great website work there if this was not just my Opera acting funny. Maybe next time NHL could pay even less for their website contractor.