Leave the wish reserved. That's what stops anyone buying a duplicate. There's no "bought" button for gifters, because Wishpicks has no way of telling whether an order actually went through at the store. Then arrange delivery, either at the store's checkout or with the owner directly.
Steps
Keep your reservation in place. Don't cancel it.
Choose shipping at the store's checkout, or ask the owner privately if you need address details.
Who marks the wish as done
Only the wishlist owner or a collaborator can mark a wish as received, and they do it once the gift is actually in hand. A reservation never does it on its own. It's the signal to other guests that this one's taken, and nothing more.
Good to know
The surprise holds. By default the owner doesn't see that anything on the list was reserved, and your note and your name stay hidden unless their settings reveal them. They do get a quiet Gift reserved notification, plus an email if they've left Gift reservations notifications on, but it names the wishlist and nothing else: not the item, not you. See "Can I see who reserved or bought a gift?" in Gift Reservation.
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 with the recipient, or ask the owner for details privately. See "How will someone know where to send my gift?"