Shiriki avoids the worst of it. He's probably not nauseous, but "slippery" might be a fair modifier. Don't worry: the vomit will freeze and chip away soon enough.
Anarchy is funny about first aid. It's mentioned as something you can spend a plot point on, but then the effect is fixed and not dependent on a roll. In fact, I'm not sure I can find a reason in the book to roll Biotech, unless the skill and the plot point are parallel systems. Perhaps you can use plot points independently and whenever you might need one, rather than burning a Narration on first aid.
Shiriki breaks out his medicine bag and finds Ye Olde Trauma Patch or Stabilization preparation or whatever rednblack sees fit to find. He manages to find a patch of exposed skin to apply it to. Ité will not immediately expire, although it might have been more convenient if he had.
So, first aid is weird. It's listed as something that falls under the Biotech skill, but no real rules exist for it that I could find. And then there's the Plot Point business, which doesn't require a skill at all. It should be noted that Shiriki's Medicine Bag doesn't have medical supplies, and so far it's purely an RP item. I was operating under the assumption that we all had at least a basic medkit in tow.
About taking actions to keep Ite alive, I've been trying to figure out how to proceed here. Shiriki and the rest of the team wouldn't ICly leave him, or take actions which would allow him to expire, but with Pap gone that puts us in a bit of a bind.
I think our best bet is for Shiriki to exit the copse of trees, take the pilot's seat and physically jack in to the Dragon -- no chance of enemy hacking at that point -- and have 2 of the stronger team members load Ite onto the Dragon before departing.