I’m guessing that it will be a 27 mhz set, in which case a new transmitter is possible, but difficult to find new, it’s oudated technology. However, you need to be sure. Have a look at the electronics inside, see if any numbers are showing.
Is there no info on the transmitter? Or nothing in the instructions? Perhaps a web search for whatever model you’ve got will help. See what you can find out.
Do you have any electrical skills? Or know anyone who has. Changing a switch is usualy a simple job.