So you want to put a wireless transmitter outside the gate, so visitors can push that button, and it will trigger a receiver that will activate the wired doorbell in your house?
You can use any generic radio controlled switch for this. They are available fairly cheaply on eBay - search for wireless relay.
They consist of a packaged transmitter with one or more pushbuttons - probably not waterproof or weatherproof - and a circuit board with one or more relays. You need to supply a voltage to the board - a "wall wart" is fine for this. You connect the relay contacts across one of the doorbell pushbuttons, so while the relay is closed, the doorbell system sees a doorbell button being held in.
The wireless relay system has to transfer the pushbutton state to the relay state. In other words, while the transmitter button is held in, the relay remains closed; when the button is released, the relay contacts open. Some wireless relay systems have different behaviours or are configurable. For example, each press of the button toggles the relay from ON to OFF or vice versa. Make sure you get one that controls the relay state directly from the pushbutton (or can be configured to do so).
Edit: Oh, and welcome to Electronics Point