No, rings opened in website opener not makes into HOF board, it is a fact we (almost) all know.
But if you want evidence, see post here with photo about ring opened at website:
https://www.planetcalypsoforum.com/forum/index.php?threads/hw-2021-ring.284382/#post-3785802
Now go to EL and search the rare loot for halloween ring 2021, you wont see Sean Rocket Connors there, because it was from website, no HOF ingame, not made it into HOF board. It is same for all rings, not just seasonal. I would assume it is same for rare tokens too.
I guess it would make into hof board only if its the first, as it would give the discovery hof at login or when looted at website, depends when the system places it into your inventory, guess at login, based on Sophet pet's from the past.