To let friends reserve gifts, make sure your wishlist is viewable (it is by default) and share the link. Guests can then reserve gifts — by default even without an account.
Steps
Keep the wishlist set to Anyone with the link or Public so people you share with can open it. An Invite only wishlist only lets people you've invited by email reserve. See "How do wishlist privacy settings work?"
Check your reservation setting under the Reservations tab in your wishlist settings. By default it's set to Anonymous Reservations, so anyone with the link can reserve. If you switch it to Registered User Reservations, friends must sign in first before they can reserve.
Share the link. See "How do I send my wishlist to someone?"
Your friends open the wishlist, tap Reserve on a gift, and confirm.
What to expect
Reserving locks the item — one reservation per gift. So if a friend sees "already reserved by someone else," someone else claimed it first.
Two settings shape what you see afterwards:
By default you won't see that a gift was reserved — that keeps the surprise. Tap Reveal reservations (in the wishlist's filters bar) if you want to see what's been claimed. You'll get a "Surprise Spoiler Alert!" prompt first, and the button then switches to Hide reservations.
Whether you can see who reserved depends on your Reservations setting. The default, Anonymous Reservations, hides identities; Registered User Reservations shows the reserver's name (and requires them to be signed in). See "How does gift reservation work?"
If a friend still can't reserve, check that the wishlist isn't private without inviting them.