Buy it on your account and put their address in at checkout. The order stays yours, which is what you want — the tracking, the return route and the protections all follow the account rather than the doorstep.
Why not use their account
You cannot pay for a win on somebody else's account, and you should not share a login to try. Can I pay for someone else's win?
And you would not want to: if the item is wrong, the person who can put it right is the account holder. Better that it is you.
Will they see the price?
A per-order packing slip leaves the prices off, so a parcel can travel as a gift without an invoice in it. That is the default for a single order.
If you want to be certain, ask the store — a one-line message when you buy is enough, and most will also leave off anything else you ask them to.
The one thing to check before you bid
When it will actually arrive. Two numbers, not one: the store's dispatch time, and then the courier's journey.
An auction you win on Friday for a birthday on Sunday is a bad plan unless the store posts same-day and you are nearby. Will my order go out today?
For anything with a date attached, buy earlier than feels necessary, or buy at a fixed price rather than waiting for an auction to end. Buy now, explained
What makes a good gift here
One of a kind things. That is what this marketplace is full of, and it is the opposite of what a chain shop can offer.
Something you can ask about. You can have a seller turn it over on camera before you commit, which is more than you get buying a gift anywhere else. How to ask a seller about an item, live
Something handmade, where the maker can tell you about it and you can pass that on.
If it goes wrong
Your order, your return. The window is 7 days from delivery for anything damaged, faulty, wrong or not as described, whatever the store's policy says — and it runs from when it reached them, so do not leave it unopened on a shelf until the day. How returns work
A note is worth more than packaging
Ask the store to include one, or write to them and say it is a gift. Most will. It costs nothing and it is the part people remember. Should I put a note in the parcel?
More: can I change my address, will my order go out today, how returns work.