From the Xbox Live Family Pack FAQ:
Q: Could a group of friends or roommates share an Xbox LIVE Gold Family Pack?
A: The Xbox LIVE Gold Family Pack is really designed for families. Only the primary account holder may make purchases against their own billing account, which is designed to help families better manage their entertainment budget. The primary account holder will also receive Activity Monitoring reports, and have the ability to access online Family Settings for other members in the Family Pack. As a result, the Xbox LIVE Gold Family Pack may not work well for groups of friends or roommates.
So no, they cannot purchase points through the Xbox from the Microsoft store. However, they may purchase a physical points card or points card code from a retailer like Amazon.com, Gamestop, or the Microsoft store (note that one the Microsoft store, there may be permissions that need to be enabled if it is a child account. You'll need to check on that).
Q: If my child receives a Microsoft Points card as a gift and redeems it on the console, who do the Points belong to?
A: The Microsoft Points would be credited directly to the account of the person redeeming the Microsoft Points card, in this case your child, as opposed to the primary account holder’s account. If your child were then to purchase a movie or Xbox LIVE Arcade game with his or her points, they would be subtracted from their individual account. Don’t forget that the primary account holder has the option of purchasing and giving Microsoft Points Allowances to other Family Pack members as well.
Hope this clarifies what you were asking.