Is it possible to send a gift anonymously or at least without them knowing beforehand?
I tried sending myself a gift just now - the email contains the sender's email address, indeed. What you can do is send a gift to your own secondary email address, and send them the URL for redeeming the gift (the URL you receive by emailing the gift to yourself) . Hopefully you have some way of contacting them other than their steam profile; for example, give them the url written on a slip of paper.
Caution: The URL contains the email you sent the gift to. If you don't want the receiver to be able to guess who you are, and the secondary email you're sending to is obviously related to you, create a new email account to send to. Example URL, email highlighted:
https://store.steampowered.com/account/ackgift/1F631467BB7E?redeemer=Fadeway%40yahoo.com
With this method, the first time they learn the sender's name will be when they enter the URL into a browser:
If there's no way to transmit the URL to them without posting on their steam profile, that's a lost cause, unless you get a third person to act as an arbiter/organizer who friends everybody and distributes the links while keeping it a secret who sent which one.
Here's the article about gifting: https://support.steampowered.com/kb_article.php?ref=6262-QXCN-0755
Unfortunately, you can't send a gift anonymously.
No matter how you send it from your account, your recipient will see your Steam profile name and/or the e-mail address that your account was registered with. Even if your change your community profile name, they can look at the previous names you've used.
The information below was correct at the time of writing, unfortunately it is no longer true. Gifts purchased now can no longer be stored in inventory or re-gifted onwards.
If this is part of a secret santa then presumablly there is some kind of organiser who knows the pairings.
If the orgnaiser has a steam account then as far as I can tell you should be able to gift the game to them and they should be able to receive it into their inventory (NOT their library) and gift it onwards to the final receipiant.