wireless on/off stopwatch

possiable but!

I am not the best when it comes to RF circuits.
the DTMF is no problem.
Also how fast do you want this?
If you can't locate someone else then maybe.

Why do I bother?

I said IF

you couldn't find someone more qualified
my suggestion is use a 2 way radio then connect a DTMF module to them.
If you can't find someone then yes I will build you a unit.
the easiest circuit (parts location?) is

radio remote control RC using DTMF circuit diagram

Another thought= RC transmitter/ receiver

Model airplan/car/etc transmitters etc might be the easy way to go.
Tried locating DTMF vhips etc but really no luck.
Then it hit me- why reinvent the wheel?
Get a simple 1 or 2 channel RC transmitter receiver then attach a micro switch or ? to the start gate then when the gate opens the transmitter sends a signal to the receiver and BAM you start the stopwatch.
couldn't be simpler.
Look on Ebay for used transmitter/ receiver sets