No. You can't buy a gift on Wishpicks itself. It doesn't sell products or take payment for gifts. Each wish links out to a store, and you buy the item there.
There's no cart and no checkout for gifts here. A wish is a saved gift idea with a link to the shop the owner chose.
Steps
Open the wishlist link the owner shared with you.
Find the wish you'd like to give.
Use the wish's Buy link to open the store.
Buy the item there, paying and choosing delivery at the store's checkout.
Wishpicks is a wishlist platform, not a shop. It saves gift ideas and links to stores, but sells nothing itself, and no money passes through it. See "Where do I buy the product?" and "Does Wishpicks process payments?"
Good to know
Reserve first so nobody double-buys. Reserving marks a wish as taken, but it doesn't buy the item. See "Do I have to reserve a gift before I buy it?"
The one paid feature is Wishpicks Plus, an optional subscription for price-tracking tools. It has nothing to do with buying gifts.
There's no delivery-address field on a wish, wishlist, or profile. You arrange delivery with the recipient, or at the store's checkout. See "How will someone know where to send my gift?"