1.) Raids
There should be an option to be able to raid a random vault of another player, you would send however many dwellers you feel is necessary and make a journey to another vault, the trip would take maybe 10 minutes and in that you would face any dangers as you would scavanging, so not only would you have to face another players army but the outside world. It would work just as a raider invasion but you would be able to steal loot as well, any dwellers who die would be permanently dead, no revives so it's a risky maneuver to take. There would be a general skill matching so powerful players won't invade new bases and vice versa.
2.)Auction house/Dweller trade
Basically just a simple market place to trade loot or dwellers for caps, have 50 or so random vaults cycle each day so the auction isn't flooded with every item.
3.) Scavenging interactions
When you send out a dweller there should be a small chance of running into another players dweller and you can have the options of trading, ignoring or attacking, you could set your dweller to a specific behavior so they do that every time they meet someone, when you meet someone a little notification would pop up and you'd click on it and goto the two players, if attacking it would be just like attacking a raider except significantly harder, you cold use the stimpaks on command when in battle to make it more interesting. When one is victorious they would have the option to take all the loot and caps. If a dweller dies from another dweller they are dead permanently, I think it would give a nice risk and reward system to the game, if a player is returning then no interactions would occur like normal.
4.)Alliances/enemies
Be able to form an alliance with other vaults to be able to share resources and loot with each other, have a system set up to make enemies of the alliance so when you raid them or kill one in the wastes you get bonus xp