I'm pretty sure they're not woodworkers.
"Chippendale" is the word's most famous cabinet maker (some people know his name, others don't, it's entirely likely that children watching the cartoons didn't know the name, but their parents likely did).
"Chip" is a perfectly good name for a chipmunk. But when the second chipmunk is "and Dale", the second name doesn't make as much sense until you realize that it's "Chip and Dale" or "Chippendale".
It's like if you have a ninja who's name is "Nin", and their partner is named "Tendo". The names sort of work on their own, but then you realize that the writer was trying to make a "Nintendo" pun. The two ninjas don't need to play videogames, and some might say the pun would be too forced if they did play videogames.
(BTW, "Chippendales" is also the name of a group of male strippers who rose to fame later, which changed the context of the "Chip and Dale" naming pun.)