I am having problems with the chip constantly resetting. Can this be due to the 120mA output from the regulator? Or is it most likely in my program??
It's hard to say. If you are drawing a lot of current or have large current surges, such as when controlling a servo or solenoid, you may be overheating the regulator, causing a collapse of the supply voltage or it to going into thermal shut down, which resets the PICAXE.
Most simple PICAXE circuits, even with a few LED's and other bits and pieces, shouldn't draw too much current, so should not cause any reset.
It could be that there's a circuit error which causes a large current surge when the PICAXE does something with I/O pins which causes a current surge, shuts down the regulator, which causes a reset.
If you have something connected to the Download Serial In pin beyond the download cable; that can cause resets unless a DISCONNECT command has been used at the start of the program.
Another possibility could be that you have a RETURN somewhere which gets executed without a corresponding GOSUB which causes the program to go off in a random direction, which may be likely to give the impression the PICAXE has reset.
The key questions would be; how frequently the resets occur, what you have connected to the PICAXE, and what your program does. Seeing a circuit diagram and the program code may help diagnose the issue.