After you buy the gift in the store, just leave the wish reserved so no one buys a duplicate. There's no "bought" button for gifters — Wishpicks can't confirm an external-store purchase, and only the wishlist owner can later mark an item as received. Then arrange delivery through the store or directly with the owner.
Steps
Keep your reservation in place. Don't cancel it — that's what stops anyone buying the same item.
Arrange delivery. Choose shipping at the store's checkout, or coordinate with the owner privately if you need address details.
Done. You don't need to mark anything as "bought" — there's no such action for gifters.
Why it works this way
Wishpicks can show that a gift was reserved, but it has no way of knowing whether you actually completed a purchase in the external store. That's why there's no "bought" or "purchased" status for gifters: your reservation is the signal to others that the item is taken. Marking the wish as fulfilled comes later and belongs to the owner or a collaborator, once they have the gift in hand — a reservation never does it on its own.
The surprise is preserved. By default the owner doesn't see reservations, and your reservation note and identity stay hidden unless their settings reveal them. The owner may receive an in-app notification, and — if they've kept their Reservation alerts setting on — an email letting them know a gift was reserved, but reservations stay hidden until they choose to reveal them, so the surprise is protected. See "Can I see who reserved or bought a gift?" in Gift Reservation.
Need to back out? If you can no longer give the gift, cancel your reservation so it frees up for someone else. See "How to cancel a reservation" in Gift Reservation.
There's no delivery address field anywhere on Wishpicks. You buy at the store and arrange delivery directly with the recipient, or ask the owner for details privately. See "How will someone know where to send my gift?"