Power ATX motherboard from JAMMA
You can, provided you know the amount of current your motherboard requires, and that the power supply in your arcade cabinet can supply that amount of current.

Furthermore, there is more going on there then I think you think. The ATX standard has 3.3V, 5V, 12V, -12V and sometimes -5V connections:  



You would also need to fake the PWR_ON/PWR_OK signals, which shouldn't be too hard (just a resistor pulling the relevant IO lines into the relevant states), though managing power-up and power-down sequencing could be somewhat involved (ideally, you'd want to switch the JAMMA power supply using the signal from the PS_ON line, but that would also require having a working 5V standby source.