How do modern avionics work without vacuum pumps or gyroscopes? I'm going to be taking my checkride soon on a Skycatcher 162. It has an all glass cockpit utilizing the Garmin G300, and its my understanding that it does not utilize gyroscopes or vacuum pumps for any of its instruments.

Seeing that if I were in something like a Cessna 172L, which uses both gyros and vacuum pumps for many of its instruments, questions such as "how does the attitude indicator work?" are fair game, I assume that the same questions are fair game for avionics like the Garmin G300.

The problem, I don't know how they work and its looking like the top google result for "garmin g300 gyroscope" returns a 404 error.

Also, do digital pitot/static systems work in essentially the same way as the old school ones in planes like the C172L? I'd hate to tell my examiner anything along the lines of "I have no idea".
If you haven't already done so, you should have a look at section 7 of the C162 POH. That section gives an overview of all the aircraft's systems and the information is at a good level of detail for a checkride (at least at the private level). The examiner isn't expecting you to explain how a ring laser gyro works, but he probably will expect you to be able to give an explanation like this (from the POH):


  AIR DATA, ATTITUDE AND HEADING REFERENCE SYSTEM (ADAHRS) AND MAGNETOMETER (GRS)  
  
  The ADAHRS provides airplane attitude and flight characteristics
  information to the G300 displays integrated avionics units. The ADAHRS
  unit, located behind the instrument panel, contains accelerometers,
  tilt sensors and rate sensors that replace spinning mass gyros used in
  other airplanes. The magnetometer, located in the tailcone, interfaces
  with the ADAHRS to provide heading information.
  
  The air data portion of the ADAHARS compiles information from the
  airplane's pitot-static system to calculate pressure altitude,
  airspeed, true airspeed, vertical speed and outside air temperature.
  An outside air temperature probe, mounted on top of the cabin, is
  connected to the ADAHRS.


The G300 Pilot's Guide goes into more detail about the system's components (see Section 1), although it's still at a fairly high level.