Disregard where previously I thought level shifting as the the number one concern, as I have a new one here ...
I've previously used 7805s in an application where its input was 12V. This limited the max current output at 5V to about 110mA. You're using 9V, so the max current will be somewhat higher, but I suspect you're hitting the thermal shutdown limit and the partial shutdown at 5V isn't being picked up by the multi-meter (an oscilloscope would reveal this).
- try reducing the 9V input of the 7805 to about 7.8V by connecting two 1A diodes (i.e. 1N4004 to 1N4007) in series and add a 1µ to 10µF capacitor from 7.8V to GND. Having 7.8V at the input will boost the 7805's max current substantially as opposed to using 9V.
- try using a 7.5V dc power supply for the input to the 7805
- try using a heat-sink on the 7805