No. Wishpicks isn't a shop and takes no payment for gifts: open the wish, tap Buy on [shop] (To store on a phone) and pay in the shop.
The only thing you can pay Wishpicks for is Wishpicks Plus. There's no cart or checkout for gifts, so your card details go to the shop. Wondering whether you pay extra, and how Wishpicks earns from some links? See "Is Wishpicks free, and what does Plus add?".
Steps
On a phone:
Open the list and tap the wish.
Tap To store in the bar at the bottom of the screen.
Order and pay in the shop, the way you'd buy anything online.
On a computer:
Open the list and click the wish.
Click Buy on [shop], or To store when the shop isn't named.
Order and pay in the shop.
Just reserved it? The Reserved window has the same button.
Why is there no store button?
Either you're looking at the card, or the wish has no link. A card shows the shop's web address but no button, so open the wish.
A wish with no link has no button anywhere. Buy it wherever you like; the owner's note may say where. If you reserve it, the Reserved window reminds you: "Reserving doesn't buy the gift — you'll need to do that separately."
I've bought it. Do I need to mark it as bought?
No. Buying in the shop changes nothing on Wishpicks, so keep your reservation: it tells everyone else the gift is taken. The owner, or someone helping run the list, marks it as received once they have it.
Can a few of us chip in for one gift?
Not through Wishpicks: there's no group gift or split payment, and a reservation has no amount attached. Agree who reserves and orders it, then settle the cost between you. The Contributor role on a list is about editing it, not money: see "What can each access level do?".
Do I have to reserve before buying?
No, but it's worth reserving first. Without a reservation, nobody else knows you've got it, and someone may buy the same thing: see "How do I reserve a gift?".
Tips
Read the wish before you order: the owner's note may name a size, colour or model.
Sold out at the shop, or not quite what the wish shows? Ask the owner or pick another wish.
Leave time for delivery, and find out where to send it before you check out: see "Does Wishpicks deliver the gift?".
Can't find the wish again when you're ready to buy? See "Where can I see the gifts I've reserved?".