A Rottweiler I found in fallout shelter has disappeared
Solution 1:
The things you find in quests do not show up until the team has returned.
Since pets do not show up in the party inventory, you can't tell which team has it until they have returned.
This also applies to any boxes you find. They are attached to the team, and do not show up until they return.
I think Nuka-Colas are the exception, since they go straight into your stash when you find them in the quest.