What OP AMP spec should I be checking to assure this behavior
You need to look for "rail-to-rail" op-amps and even then there will only be a few that will do 20mV (with a 1mA sink current) such as the AD8605 - not cheap but pretty good all round and beats the LM324 into pulp on most things providing it runs from less than 6V.

Here is linear technology's search engine with a few parameters also selected to give you a few to look thru

Going back to your idea of using an oscillator and small charge pump try looking for a single schmitt trigger in an SOT23 package - TI and Fairchild both make them. There are ones at TI that can output 30mA and can oscillate at tens of MHz. A couple of 1N4148 diodes and smallish capacitors should give you the negative rail you need.



Why can't one op-amp in a quad package generate a negative voltage (via a charge pump) to power the quad package's negative rail. If each op-amp takes 1mA unloaded then one op-amp has to supply 4mA but, in supplying 4mA this one particular op-amp now takes 5mA and therefore the total supply demand is now 8mA and of course this spirals out of control. On the other hand, one op-amp can be an oscillator and this oscillator can be buffered by a transistor that takes power from the positive rail independently of the op-amp. This will work so maybe consider that.

On a slightly different (but related) note, you can add an NPN BJT as an emitter follower to the output of an op-amp and use feedback from the emitter. What this gives you is worth trying. For a start you need an emitter resistor that won't drop more than 20mV when a 10k is pulling it up to 5V (a current of 0.5mA) therefore the emitter resistor can be no-more than 40 ohms. The other nice thing about this (if you can live with the extra current) is that the op-amp only needs to lower its output to ~0.7V to get ~0V at the output. This means the LM324 is now easily going to manage this situation. Here is an example: -



So, it's the 40 ohm resistor (Rload) that pulls down towards 0V to give you the low end of what you need. Going back to the original neg voltage generator, now that the op-amp is buffered from supplying its own current this method can be used to feed a charge pump.