There are many ways that it could be done. You could use a PIC, or do it in hardware.
A simple way to generate a random number is to have a counter running continuously and, the counter is stopped when a button is pushed (or in your case, when the vehicle is detected).
You need 3 outputs, so you would need a modulo 3 counter, ie. it counts 0, 1, 2, 0, 1, 2 0, etc.
A simple hardware solution would be to use the CMOS counter 4017.
This will operate with a supply voltage in the range 3 to 15 Volt. So 9 Volt would be a good choice.
To make it count in modulo 3, connect output 3 to the reset input.
Thus, when it counts from 2 to 3, it will be reset immediately.
You will need a CMOS oscillator set to run at a frequency in excess of 1kHz. The actual frequency is not critical. I suggest you use a Schmitt Trigger package such as a 40106 or 74C14, etc. for the oscillator.
Here is one that could be modified.
https://www.electro-tech-online.com/threads/need-help-with-a-little-project.15736/
Alternatively, if you want to use a PIC, then what I described above can be done in programme with some minimal associated hardware.
Let us know which you prefer (ie. hardware or PIC) so we can help further.