Any team member can add credit. Switch to the team, then open Credit Balance, then Add to Credit Balance. The page says "This deposit will be added to the team wallet.", and the member who adds it pays with their own card.
Automatic Payment: only team admins and the organization owner can set it up, change it or turn it off for the team wallet. Other members can see the settings, with the note "Only team admins and the organization owner can change auto-pay for the team wallet."
Whose card: automatic top-ups charge the saved card of the admin who last saved the team's auto-pay settings. To use a different admin's card, that admin saves the settings again.
Transfer Credit and coupons for the team are also limited to admins and the owner (see Transfer Credit and team coupons).
Details: How do organizations and teams work? and Team Billing & Spending.