Can other players try to stop you from completing Joining the Winning Team?
There's no direct way people could try to stop you. They can't attack you; there's nothing in the rules about players attacking players.

There are at least a few ways they could try to indirectly stop you:


They can try to close gates and kill monsters before you can get to them, so that you won't have any trophies you could use to buy allies.
They can deliberately leave monsters that'd block your path alive.
They could try to equip themselves for the final battle and let the ancient one wake up quickly, giving you no time for your mission. The gate limit makes it pretty easy to let the ancient one wake up early, if you're feeling daring.
If you do get close, they can try to make sure you can't get lost in time and space. If you try to hop in a gate that someone's about to close; they can decline to close those gates. So you'll be trying to get to 0 sanity/stamina in another world. To prevent that, they could also try to prioritize closing the gates to more dangerous other worlds before you're ready to get lost in time and space, so that when you get to that point, you'll probably just waste a few turns passing through. I suppose they could even try to heal you if they have the right character or item!