The transmitter (large box with handles and buttons) is a radio transmitter and you need the receiver to use it as intended.
However, if you had taken your time to download the documentation for it, on the page I linked to, you would have seen this at page 10:
Sometimes for diagnostic purposes, it may be useful to plug the transmitter
directly into the Microcontroller, and bypass the need for the RF Receiver.
This would help you determine, for instance, whether radio interference is at
fault for your robot behaving strangely, or if some other factor is to blame.
In order to use the tether feature, you will need to purchase a cable. Any telephone
handset extension cable will work (the coiled cable that goes from the
handset to the base of a corded phone, not the cable that goes from the phone
to the wall).
Now go download the manual (called "Inventor's Guide - Control") and read on. You really need to know it by heart, as it will answer most questions you may have about it.
Had you taken your research even a bit further, less than 3 minutes with Google could have given you this:
The tether port on the back of the transmitter has only four wires. From the bottom to the top they are: RF disable, Ground (battery minus), pulse width modulation out, and the fourth wire (at the top) is not connected.
Shorting the RF disable pin to ground will turn off the RF output. I presume this is used to "disable" the VEX bot as its behavior is similar to turning off the transmitter manually.
The pulse width modulation output requires a pull-up resistor to see the signal on an oscilloscope. If you don't know what a pull-up resistor does or how to set one up, don't mess with any of the transmitter or receiver stuff as you will likely fry something.
There is no way to send info to the transmitter tether port and have it transmitted to the receiver module.
Now you should be ready to build a bunch of software routines to emulate a cheap (and easy used) wired interface with an expensive R/C transmitter that codes all on a single signal line - I'll never get why though, as I would either get a receiver, or use a simpler interface (or use a single line interface with lots more channels)... That is, if I wanted to make a ROV, but I like robots better, as a ROV is essentially just a dumb R/C car and don't even need a "brain"
If I was in your position, I'd write (or call) Vex Robotics and ask nicely if you could buy a replacement receiver.