Materials required to convert NiMH rechargeable 24V/6Ah circuit (20 C-cell) to Li-Ion rechargeable circuit
Li-ion batteries put out 3.7V per cell, while NiMH generate 1.2V. So, 20 cells at 1.2 V generate 24V. With Li-ion you will need 24/3.7 = 6.5(!) cells. This is somewhat difficult to solve... Maybe the chair can accept 7 cells? That would be the equivalent of 25.9V - a bit much, though 8% might be acceptable.

If you have the space to fit more cells - maybe the same quantity as before (20 cells), you could include a switching converter to lower the voltage to the original 24V, and make better use of the space, and have more capacity. 

The pack would then have 20*3.7 = 74V, which should be convertible to 24V. But you would need to look for a charger which works at 74V which probably would not be so easy to find commercially. There are plenty of circuits for chargers around. Don't skimp - find a good circuit to avoid overheating problems!

You could also think about two sets of 7 in parallel - or 3 if you can fit the extra cell. You will have 21 cells total. More info on paralleling packs is available here.

The BQ2013H IC on the board is a controller which is for high-discharge battery packs, but is only for NiCd, NiMH or lead acid packs. It doesn't seem to support Li-Ion packs. So there is some work to be done there.

I looked up equivalents for Li-Ion, and Texas Instruments has a new one (bq34x100) which shows promise. You'll still have to redesign the board though.