Could this sentence also mean "We don't want to grow," as in grow our business. In other words, we want our business to remain small and uncomplicated.
If "We do not want to grow up." is the most correct acceptable translation, then "We don't want to age." should be an acceptable translation as well.
I don't think that has quite the same meaning. Anyone can age regardless of whether they are young or old. But growing up requires starting at the point of not being grown up, so it has a more precise meaning to say you don't want to grow up, ie become less childlike, than that you don't want to age ie gain additional years since birth or the effects of doing so like white hair and wrinkles.
When I first came across this sentence I used the reflexive "ci" (Non ci vogliamo crescere) to suggest growing up (i.e. growing ourselves). DL marked it as wrong. I cannot find a dictionary reference to 'crescere' which implies 'growing up' rather than growing something else.