Keep the wishlist viewable, which it is by default, and share the link. Guests can reserve from there, and by default they don't even need an account.
Steps
Keep the wishlist set to Anyone with the link or Public so the people you share with can open it. On an Invite only wishlist, only people you've invited by email can reserve. See "How do wishlist privacy settings work?"
Check the Reservations tab in your wishlist settings. It starts on Anonymous Reservations, which lets anyone with the link reserve. Switch it to Registered User Reservations and friends have to sign in first.
Share the link. See "How do I send my wishlist to someone?"
Your friends open the wishlist, tap Reserve on a gift, and confirm.
Good to know
One gift, one reservation. If a friend sees "already reserved by someone else," somebody beat them to it.
Two settings decide what you see afterwards. By default you won't see that anything was reserved at all, which is the point: the surprise holds. Tap Reveal reservations in the wishlist's filters bar if you'd rather know. You'll get a "Surprise Spoiler Alert!" prompt first, and the button then flips to Hide reservations.
Whether you can see who reserved comes down to that Reservations setting. Anonymous Reservations hides names. Registered User Reservations shows them, and requires the reserver to be signed in. See "How does gift reservation work?"
If a friend still can't reserve, check that the wishlist isn't set to Invite only without them on the list.