Hero,
Even with "rail-to-rail" op amps you will not get to "0.00V". Getting a descending ramp to 0.00 V has been the major challenge.
You get as near as damn it 0V, at least to two decimal places.
Assuming the signal generator's input looks like a 10k resistor connected to 0V, zero voltage won't be the problem, 5V will, so run the op-amps from a higher voltage e.g. a 9V battery would solve the problem as well as eliminate the need for rail-to-rail op-amps.
Why not have a digital solution?
If you use enough bits and an anti aliasing filter the steps won't be noticeable, a 10-bit DAC will give <5mV resolution.
I'd use a CD4040, an R2R DAC and a CD4069.
The 4040 is 14-bit counter which will give a step of just over 305µV
Two of the gates from the CD4069 would be used as a clock for the C4040, the remainder would be configured as a linear amplifier with unity gain both buffer and invert the signal and should have a low enough impedance.
If the CD4069 isn't linear enough, I'd use a dual rail-to-rail op-amp IC with one configured as a Schmitt trigger oscillator and the other as an inverting buffer.
A diode AND gate could disable the counter when finished.