I think you are thinking in the right direction though I don't know if this alone would suffice.
It would make exploring still the same old same, just with teams of four (give or take). The small bit of story wouldn't make that much of a difference and I'm assuming the outcome would probably be the same as sending 4 solo explorers out into the waste for a similar amount of time.
I really do like the idea of story-lines and think with other features this team-based questing would do wonders for the game. Other aspects to enhance this team-based exploration could be (1) quest prompts, (2) dweller traits/perks/karma, (3) revamp on dweller SPECIAL, and (4) vault archetypes.
1) Quest prompts - Once the team 'arrives' at the location of the quest after traveling X time, you would select the team and press 'start quest'. From there you'd get a few prompts asking you to choose paths you'd like them to take which would affect karma and interactions in the rest of the wasteland. Also traits/perks would help you in these encounters.
2 & 3) Dwellers are all the same, especially at the end game. I vote to remove the training rooms and go with the classic character creation (I discuss it in my first point here: http://www.gamesas.com/topic/1534394-improvements-to-fos-forward-looking-to-fos-2/)where you give the GOAT exam to dwellers and choose their SPECIAL along with two optional traits and you'd gain perks throughout the levelling process. This would help create a barrier so you wouldn't be able to just max 4 super human dwellers in their SPECIAL and send them out. You'd have to put a little thought into crafting your team. Karma would also play a big role in incidents/raids on your vault along with overall how your dwellers are perceived in the wastelands whilst exploring. Add layers of depth.
4) Gemini Engine talks about this in his post (which I credit in the above link, go there for the link) archetypes such as a vault with all Super Mutants or Ghouls. These vaults would have vastly different story/quest lines which would add more depth and a bunch of replay value.