You can assemble a solar charger, but it would take a very serious solar panel to power it! You're looking at something that would be larger and more expensive than the scooters. If you try to run the existing charger with a solar panel that's not supplying enough power, you will damage the panel's inverter/voltage regulator or even the charger itself.
I have solar charging on my RV, and a moderately small panel that's just enough to keep the batteries full while it's parked is over 4 square feet and cost $700 for the panel and voltage regulator.
If you want portable charging ability without getting into any custom equipment, the best thing would be to get a deep cycle battery (boat supply store) and inverter. This will run the scooter's existing charger. This will retain the speed, performance, and safety of the charger that came with the scooter.
Project Cost:
A good deep cycle battery (Interstate HD-24DP HIGHLY RECOMMENDED) - $90-$110
A cheapo (Coleman, ETC.) 300w inverter - $30-$40
A decent automatic 10-20 watt recharger for deep cycle battery - $50-$60
Now just charge up the deep cycle and take it with you wherever you need AC power. You can run more than just the scooter charger off of the inverter. You might end up getting a larger inverter if you want to run more or larger appliances, but 300watt will let you run/charge your laptop, phone, portable stereo, etc.
OPTION 2:
If you want to assemble a smaller, cheaper solar charger for the scooters, you will first need to select your solar panel. Get the best one you can,and keep in mind that sone of the "high output" models are crap in moringin/evening/shade, while some of the moderate ones are better in non-perfect conditions.
Next, you will need a voltage regulator to supply the scooter's charging voltage. Solar panels will not put out the right voltage. It will not even be constant.
You can check the output voltage the existing charger to find out what the charging voltage should be. This is NOT the same as the rated voltage of the battery. (For instance, 12v car batteries are charged at 15.5 volts.) Now, get your hands on an adjustable voltage regulator, or have a dedicated one assembled by a local electronics shop. Make sure it can handle the maximum wattage output of your panel, but don't waste money on a ridiculously oversized unit. The regulator will transform the panel's output to the proper charging voltage, but will probably at a far lower wattage (less power, fewer charging amps, longer charge time) than the old plug-in charger.
Finally, you'll also need to make sure that there's a shutoff device to interrupt or attenuate the charging circuit before overcharging occurs. If you like to live dangerously, you chould just check the charge level periodically.