"Family and community-based" is fine. I don't know of a rule, but this kind of coordinated compounding often works with various hyphenated bases: There are local and nationally-run committees. It's a mineral and oil-rich country. These are tax and VAT-free goods. And it's not restricted to compound adjectives. These hyphenated noun compounds work well too: Ed is a window and gutter-cleaner. Kim is a keen cinema and theatre-goer. It's suitable for both country and town-dwellers. But multiple compounding of non-hyphenated words does not work well: *This is a swim and sportswear shop. *There are black and bluebirds in the area. *I'm suffering from back and toothache. PaulM
For the APA style guide, at least, you don't hyphenate compound adjectives if one ends in -ly. I.e., nationally-run would just be nationally run.