Powering Motors in War Robot
First, you don't want to use that driver. Your motor spec of 12V/7.5 A assumes the motor is running "normally", and for a BattleBots situation you can't count on that: if your bot gets into a shoving match with another you'll need to deal with the stall torque, which is much higher. Since the controller you're looking at is only good for 5 amps continuous (and that probably assumes good ventilation, which I'll bet you're not providing), it will probably overheat and die.

Second, if you provide a 12 volt battery and get about 12 volts max to the motor, 45 minutes is 3/4 of an hour, times 7.5 amps is about 6 amp-hours. This is perfectly reasonable from a rather small lead-acid battery. Again, though you need to look at the possibility of larger load currents and more power from the battery. 

So. Get the data sheet for the motor and find the stall current. Alternatively, measure the motor winding resistance and apply this to 12 volts for worst-case current. Even better, get a motor and measure your currents.

If you have gearmotors, you need to look closely at your gear ratios. 200 rpm (I assume that's output rpm) is about 3.3 rps. Assuming 20 cm diameter wheels, your top speed will be on the order of 3.3 x 60 cm/sec, or about 2 m/sec - about 1.3 mph if you don't like metric. That is a very slow vehicle. 

Finally, 4 motors suggests that you really don't know what you're doing. Steering with two motors is standard. If you have 4 wheels with a motor each, you will not get good results using differential power. It just doesn't work well. You'll need a steering mechanism like a car, and making one of those which is resistant to shock and impact will be a real test.