IF you want to run 1 direction only you can use any transistor or MOSFET rated for more than 1 amp. But if you want to be able to reverse the motor you will need to make an H-Bridge and that will probably cost more than $10.
You might want to review that spec sheet again - that motor draws more than 4 amps when stalled; that is the number you want to pay attention to. So you need an h-bridge capable of supplying that, plus about 25% (so about 5 amps). For instance, this one would fit the bill, but is above (3x) your budget:
You could try to find a different motor with a lower stall current. Another option (and not a great option) would be to use an L298 based motor driver, running in bridged mode, with a -really- big heatsink on it (and maybe active cooling). In bridged mode, the L298 can supply 4 amps; your motor stalled will go over that, of course, but the thermal protection on the L298 will kick in, -hopefully- protecting the L298. I am not sure how long it takes for it to kick in, but maybe if you are careful not to stall the motor too often (or too long), it may be able (for brief periods) to allow for a slightly higher current draw.
You can easily find L298 driver boards on Ebay for well under $10.00, drop shipped from China. Most are designed to provide separate motor channels, and some don't support "bridged mode" out of the box. You may have to review the circuit schematic for the board, and reference the spec sheet for the L298 (note - there are a couple versions of the L298 datasheet floating around out there - get both!) to determine how to hook up the bridged mode (you may need to mod the PCB). You'll probably also have to install a better heatsink than the one provided, too.
I also seem to recall in my ebay wanderings running across some fairly cheap ($15-20.00 USD) h-bridge boards based on a different chipset that would allow you to source quite a few more amps than the L298 - so you might do some more research there.
If yout bot it is just going to be an RC bot, then you might want to considerr using continous rotation servos for the drive motors. Google and youtube search for terms like "RC robot" to see what others have done.