Can a wireless keyboard mouse work with a laptop's Wi-Fi rather than inserting its own USB stick?

No. A wireless mouse/keyboard does not use regular WiFi (i.e. 802.11x) and can only bind with the receiver it came with. (An exception may be the Logitech Unifying receiver, which allows connecting every Logitech device that supports it, to a single receiver - but still, it will take one USB port.)

If you do not want to use a receiver, consider using a Bluetooth mouse and/or keyboard.

Yes and no. Yes, this is possible. No, it doesn’t work with WiFi (802.11).

Instead, this can be accomplished with Bluetooth, another wireless technology built into most modern notebooks. If yours is equipped with it, you can use Bluetooth mice and keyboards. There’s plenty available.

Actually, HP makes a Wi-Fi mouse. I have one, and it works most of the time. It is called the "Wi-Fi Mobile Mouse".

As others have mentioned, Bluetooth is a better way to go.