Is there such a thing as a USB cable adapter that you can plug, say, a webcam into, and then plug the adapter into the arduino board, and then plug a radio transmitter into the same arduino board, and plug a reciver into the computer so that you can see what the webcam sees on the computer.
Would it work?
Where would you find an adapter?
Absolutely not.
The Arduino doesn't have nearly enough memory to capture video OR transmit it wirelessly.
You could MAYBE make the Arduino control something that does this, but the Arduino could not.
The short answer is "No": you need a CPU with at least 1990s-PC-level horsepower to read the video from the webcam over USB.
You could mount a regular analog video camera, and use a video modulator to transmit the signal to a TV or similar receiver. You could take that received video and run it into a capture card in a PC if you want to make a digital recording, transmit it over the internet, etc.
You can also get "IP cameras" that translate the video into internet-able packets. Some of those come with WiFi interfaces instead of/in addition to Ethernet.
What would be the role of the Arduino in this setup?
I would just use a wireless video transmitter & receiver. If you need to capture the video on the computer you could use a capture card.
If you want an IP-cam I can recommend Axis cams, they are great quality.
Remember that the Arduino is a slave USB device, not a host. To plug a slave device (the USB webcam) into the Arduino, you'd need the Arduino to be a USB host. USB host devices are much more complicated than slaves, in general.