Given that Orion has already visited Eternum while asleep, continuing the adventure while dying would make sense (after all, it's a time-limited thing, so the gang can't wait after he recovers). I don't know how this will play out, though.
- If he just logs back in, Nancy and Nova will freak out, since he's not with either of them. They will disconnect to find him and rush him to the hospital.
- If Nova realizes he's late, she might follow him to the pizza place, find him and call the ER. Then Orion finds himself in Eternum, everyone freaks out, but gets the McGuffin, use it to heal Orion and trick the Coyote to get the gem
- Or Calypso straight up drags him into Eternum, Penny heals him (hope you gave her the right tome!), they finish the quest and he steps back out through a portal.