What is the fuel source for the APU on a B737?
It comes from the left side of the fuel manifold, which means it is most often fed from the #1 main tank, typically by turning on either or both boost pumps in that tank. However, it can be fed from the center tank (using the left center tank boost pump) or from the #2 main tank using a boost pump there and opening the crossfeed valve.

If no boost pumps are running, it suction (not gravity) feeds from the #1 main tank. This would be the case when starting the APU with only battery power and no external AC power. It isn't preferrd to do this for longer than necessary, though, because it puts more wear on the fuel control unit with the suction feed than when the fuel is being supplied from a boost pump.

There is no separate fuel tank dedicated to the APU. Gravity feed wouldn't work because the APU is higher than the main tanks (in the wings) and the center tank (below the cabin floor).  (The exception to that would be if the aircraft is inverted, although that situation is considered undesirable for several reasons unrelated to providing fuel to the APU.)