Yes. Right after you reserve a wish, Wishpicks offers a box for a short note. It's optional, and skipping it is fine.
One thing to know first: the note isn't private from the owner. If they turn on Reveal reservations to see what's been claimed, your note sits next to the reservation. Treat it as a message the owner might read, and leave out anything that spoils the surprise.
Steps
Open the wishlist link the owner shared with you.
Tap Reserve on the wish you want to give and confirm.
Type your note in the message box on the confirmation screen.
Tap Leave a message to save it. To skip it, just close the screen.
Who can see your note
The owner can, the moment they turn on Reveal reservations. Other gift-givers browsing the same wishlist can tell an item is reserved, but they can't read what you wrote.
Good to know
Don't put a delivery address in the note. Wishpicks has no delivery-address feature, so arrange where to send the gift with the recipient directly, off Wishpicks.
While the confirmation is still on screen you can rework the note: Edit sits next to Message sent. Once you close that window there's no way back to it, so get the wording right before you leave. Cancelling the reservation removes the note along with it.
Related
"How does gift reservation work?"
"I reserved a gift by accident — how do I undo it?"