Can I use the MMA8452q 3 axis accelerometer without a microprocessor? If so, how? If not, what would be a good accelerometer to use that does not require a microprocessor?
From a brief glance the MMA8452q appears to be digital only. This means that the easiest way to interface with it would be to use a microcontroller.
The ADXL337 (https://www.sparkfun.com/products/12786) on the other hand is an analog sensor. It is less configurable but the acceleration measured on each axis is represented just by the voltage level on an output pin. This would be easier to use in a project that does not have/need a microcontroller – however you may need additional circuitry to amplify or respond to the signals that the accelerometer outputs.
As a side note you could actually use a digital accelerometer without a microcontroller too – but it would be hard and a purely educational experience IMO. You could use discrete logic as well as parallel to serial converters and other special ICs to create a circuit that polled the accelerometer for information.