This could also be "Would they like a shirt?"
You'd use conditional for that.
Would the translations be different for "They want" and "They would like"?
The conditional of "vogliono" is "vorrebbero", but I wouldn't use it in this case.
Do you have some specific case in mind?
No I'm not thinking of a specific case, but the dictionary hints for vogliono suggest want, would like, and wish. I entered "Would they like a shirt?" and it was wrong, but I wondered why.
I think you can use it, but my English knowledge is not perfect. You can try to suggest to use it in case you find this sentence once again. :)
I put 'Do they want a jacket' but it came out as wrong!?!?
"Camicia" means shirt.
If you used "jacket" because it was in the hover hints see here how to get the most out of the hints.
Random rant: This speaking voice on this app needs to be MUCH clearer. The difference in her open "e" and bright "a" are almost indistinguishable at times.