To mark that you'll buy a gift, open the wishlist someone shared with you, tap Reserve on the wish, and confirm — that flags the item as taken by you so no one else picks the same thing.
There's no separate "I'll buy this" button. Reserve is how you signal your intent. You then go and buy the item yourself at the store.
Steps
Open the wishlist link the owner shared with you.
Find the wish you plan to give.
Tap Reserve and confirm.
Optionally add a private note for other gift-givers (the owner never sees it).
When you're ready, tap the store link on the wish to buy the item at the merchant.
A few details
You may not need to sign in. If the owner allows open reservations (Anonymous Reservations), you can reserve as a guest without an account; if they limit reservations to registered users (Registered User Reservations), you'll be asked to sign in first.
One reservation per item. If someone has already reserved a wish, you'll see it's taken and can't reserve it until they cancel. See "Can two people reserve the same gift?"
The owner usually can't see your reservation. By default the "Reveal reservations" setting is off, so the surprise is safe. See "Can I see if someone reserved my gift?"
Reserving does not mark the wish as received. The owner marks an item as fulfilled later, on their side — the gift-giver can't do that.
Reserving and buying stay two separate things. Reserving just holds the item on Wishpicks so gift-givers don't overlap, while buying happens at the external store — Wishpicks is not a shop and doesn't take payment. There's no delivery-address feature here either, so you arrange delivery directly with the recipient, or the recipient shares their details with you privately, off-platform. See "How does gift reservation work?"