Does VSS (Logic Supply Voltage) on a L293D need to be connected for the IC to work?
The reason it's drawing large amounts of current is that this is an older part that uses TTL input stages, instead of CMOS input buffers: here's what the inside looks like, at least from TI:



There does not appear to be an ESD protection diode to clamp the input to VCC1, so that's probably not the leakage path that is allowing this to work. I think it's still working because you have some leakage path elsewhere (please post a schematic) that is supplying VCC1 -- try measuring VCC1 in your case with a DMM and see what voltage it shows. I wouldn't leave VCC1 unconnected for the actual end product -- find a switch to turn it on/off when you need too.

As an aside, you do appear to have a bunch of floating inputs; while this isn't CMOS, I would still tie off your unused inputs to known values.