Does Apple charge a fee to load an Apple Gift Card to an Apple wallet?

No. no fee is charged for Apple Gift card redemption.

But do note:

You cannot add an Apple Gift Card directly to the Apple Wallet, you can only add it to your Apple Account. you can go through the Apple Wallet for this, but the card is not added to the Apple Wallet App directly. It's balance is transfered onto your Apple Account.

click here ➜ Redeem your Apple Gift Card, App Store Card, or App Store & iTunes Gift Card - Apple Support

No.

If you have a previous purchase to your Apple Account that has not been paid, then that attempt will be made when you have added to funds to your Apple Account.

Apple gift cards can only be used to purchase Apple products and services, that’s it. So if you were hoping to be able to load the gift card into Apple Cash or other card and spend it as you wish that’s not possible. Gift cards can only be redeemed into your Apple Account (formerly called Apple ID) and can only be used on Apple products and services, nothing else.