Can I buy Steam Game Keys on Steam?

No. You, as normal user and not game developer, can't buy keys from Steam. But you can achieve the same with Steam gifts:

  1. Buy game as a gift.
  2. Send it to your e-mail address.
  3. Copy activation URL (Just https://store.steampowered.com/account/ackgift/<some numbers and letters>, completely omit redeemer=<e-mail address>).
  4. Send it to someone.

Just tested it so it works for sure.