I'd like to purchase iCloud storage plan as a family plan, but my family members have their own apple IDs. Can they back up their phones to the family storage anyway?

If you are the Family Organizer of the Family Sharing Plan, and you have subscribed to 200 GB or 2 TB of iCloud Storage then, yes, you are able to share this iCloud storage with your family members (in your Family Sharing Plan).

Each Family Member would use their own (existing) individual Apple IDs and would have access to only their own iCloud data, as if they were using their own individual iCloud Storages ... the only thing is that there is one iCloud Storage now and you, as the Family Organizer, will be billed for it.

See this to understand more and how to go about it: Share an iCloud storage plan with your family - Apple Support