Can I purchase the yearly plan of AppleCare with my Apple Gift Card?

You can probably use it for actually paying, but if this is a subscription you will need to have a "valid payment method" such as a credit or debit card to be present on the account even if it isn't actually used to make the payment in the end.

This article explains the different kinds of gift cards Apple uses and has links on how to redeem them and what can be purchased with each kind. Click on the link --> What type of gift card do I have? - What type of gift card do I have? - Apple Support

How apps, content, and subscriptions from Apple are billed - Apple Support

Apple Gift Card and Apple Account balance can't be used for some purchases

• For some recurring charges, like AppleCare+, you can't start your subscription with Apple Account balance. But as long as you keep another valid payment method on file and your Apple Account balance would cover an entire charge, your recurring charge will pull from your Apple Account balance.

About linking AppleCare to an account:

If you can't find your AppleCare plan - Apple Support

Re: "Apple care with gift card:

I have apple gift card Can i purchase the yearly plan of apple care with it? "

You may need to check what that particular gift card can be used for:

(1) See : What type of gift card do I have? - Apple Support

and

(2) See : What you can buy with your Apple Gift Card or Apple Account balance - Apple Support