No. Wishpicks is a wishlist, not an online shop. It keeps the gift ideas you and your friends want and links out to the stores that sell them. Whoever's buying pays that store directly. Wishpicks never sells the product or takes money for a gift.
How buying actually works
A wish links to the product page on an outside store.
Whoever wants to buy it clicks through to that store.
They pay and check out in the store, the way they normally would.
Wishpicks plays no part in that. There's no cart or checkout for gifts here.
Good to know
The one thing you can pay Wishpicks for is Wishpicks Plus, an optional subscription for price tracking. It lets you track prices at any store and set your own price thresholds. It doesn't let you buy gifts through Wishpicks.
Reserving isn't buying either. A gifter reserves an item so other people know it's taken, but the purchase happens at the store. That's also why Wishpicks can't confirm a purchase: checkout happens somewhere else, so Wishpicks knows an item was reserved and nothing more. See "How does gift reservation work?".
To find the shop for a particular wish, open it and follow its link. See "Where do I buy the product?".
For the bigger picture, see "What is Wishpicks and how does it work?".