Can a family member pay for a subscription on their own account

No that is not possible. See these basic reads on family sharing.

Family Sharing - Apple Support

Set up Family Sharing – Apple Support

Share App Store, iTunes Store, and Apple Books purchases with Family Sharing - Apple Support

From the third link, "With purchase sharing, one adult in your Family Sharing group agrees to pay for any purchases from the App Store, iTunes Store, and Apple Books. You can then see and download your family members' purchases from your iPhone, iPad, iPod touch, Mac, Apple TV, and PC, and enjoy them too.* Follow the steps below to download family members' purchases to your device, and learn where to find those purchases if you don't see them."

In some cases, sharing a subscription depends on if it is a family subscription or a single person subscription. See the 'What subscription plans are available' section of the article below for some examples.

Apple Music subscription - Manage.    

See the documents on family sharing. You should be able to access what he downloads. And, right you, the organizer, have to pay.