It depends upon the type of coverage you are buying. If it is a single payment you can buy it with your card. If it is a subscription you might have to set up Family Sharing with you as the Organizer so your payment method is used for any subscription purchases a family member makes. The AppleCare One would have to be taken out using the other family member's account.
Another possibility for a subscription AppleCare:
Once started, you could gift an amount required to cover the monthly cost:
From: How apps, content, and subscriptions from Apple are billed - Apple Support
For some recurring charges, like AppleCare+, you can't start a subscription with Apple Account balance. But as long as you keep another valid payment method on file and the person's Apple Account balance would cover an entire charge, and the recurring charges will pull from their Apple Account balance.* * In some countries and regions, your Apple Account balance can’t be used for some subscriptions like AppleCare+. Check if Apple Account balance in your country or region isn’t supported."
