Stumped on an inverter
I assume that your "in the wall charger" is not a charger per se, but a 24VDC power supply, and that the actual battery charging circuitry is integrated in your chair or in the battery pack in your chair.

The rest of this answer is based on the assumptions in the above paragraph. If any of them are not correct, you should disregard the answer.

However, if the assumptions are correct, the solution is potentially relatively simple.

If the chair charges from 24VDC, and your car has 12VDC, you can skip the AC (inverter) and the wall power supply (the "charger") altogether.

Use a suitable 12VDC to 24VDC boost converter. 

It will be lighter, cheaper and probably more efficient than using an inverter to convert 12VDC to 110/240VAC and then to 24VDC.

Some requirements:


The boost converter must be able to supply a stable 24VDC at 5A to the charger integrated in your chair.
Naturally, you'll need to equip it with the correct plug.
The boost converter must be able to tolerate up to at least 15VDC on the input, and possible spikes of even higher voltage, because the voltage in a car can vary quite a lot.
The car must be able to supply at least 10A at 12V without blowing a fuse or catching fire.


You should speak to an electronics and/or car professional about this. While it's not rocket surgery, there is quite a bit of energy involved. Also a potential (lithium battery?) fire hazard in a moving vehicle. All in all, you should make sure that it's done right, or not at all. Hopefully this answer can point you in the right direction.



Edit: If the wall charger is an actual multi-stage battery charger, like the one in your link, the answer above doesn't apply. But did you ask the supplyer of your charger if they also have a model that can run on 12VDC? 

The website was pretty sparse on technical details, but the batteries on the site seem to be lead-acid type batteries. There are plenty of chargers out there for 2x12V lead acid batteries, that can take a 12VDC input. 

I tried to google "24v battery charger 12v input", and found several retailers. If your usual retailer doesn't sell 12VDC driven 24V chargers for your particular type of battery, I'm sure somebodye else does. 

If you find a suitable and compatible charger, I personally feel that would be a "cleaner" solution than using an inverter. That being said, using an inverter together with your existing charger would likely work too.