My reading is that no in-universe explanation is needed (and that one would in fact be undesirable) because the character of Thing is (in the TV series) a metafictional joke: the viewing audience is intended to understand that Thing is not really a disembodied hand but is of course played by an actor, the rest of whose body must have space to be hidden.
Therefore Thing always has to pop out of a box or opening of some kind, which can be used to conceal the actor's body. And that's the joke.
Metafiction is a literary device used to self-consciously and systematically draw attention to a work's status as an artifact. It poses questions about the relationship between fiction and reality, usually using irony and self-reflection. It can be compared to presentational theatre, which does not let the audience forget it is viewing a play; metafiction forces readers to be aware that they are reading a fictional work.
So an in-universe explanation would actually break this joke, because the real-world limitation on Thing's appearances - that there must always be space for the rest of the actor's body - is the joke.
If one adds an in-universe reason why Thing always pops out of a box, or if one starts to explain how Thing moves from one place to another, then the joke is broken, because one no longer needs to take into account the Addams Family's "status as an artifact"; if there are in-universe reasons for Thing's limitations then they are no longer an ironic reference to the Addams Family's status as a fictional TV show.
This is a "breaking the fourth wall" joke; asking for an in-universe explanation is like asking for an in-universe explanation of how Ferris Bueller makes letters appear in mid-air - there isn't one, and if there were then it would be intrinsically unsatisfying:
In the Addams Family movie the makers do completely break the joke by using special effects to make Thing actually be a disembodied hand, because the movie is (compared to the TV series) tone-deaf and about as subtle as a brick. IMO.