On a 24v electric battery powered scooter using a 2 lead acid batteries as power,why cant you....?

have sum form of an alternator to recharge the batteries as you travel, as opposed to having to recharge them with a charger at your home?

a double pulley with the one pulley that powers it and the other recharging it? that could spin an alternator.

You cannot use the batteries for power and recharge them at the same exact time, but there is something called regenerative braking. When you want to stop, you use the motor as a generator and recharge the battery. This takes the energy that would be wasted with brakes and puts some of it back into the battery. You will always have to recharge the battery at home eventually (or use a solar panel). It is impossible to get back all the energy you used to get moving. Some of it is lost to air resistance, wheel bearings, heat in the motor and battery, etc.

Yeah you can, however it doesn't mean it will be efficient. In fact I honestly can't see such a system recharging the batteries enough.
Load, speed and use of scooter simply won't charge them enough to be self-sustaining, so to speak.
My 2cents worth.

What would power the alternator?

I don't think you could drive the alternator fast enough to keep the battery charged.

It's a perpetual motion machine. It won't work.

You will use up, say 1000 watt-hours driving the alternator to recover 500 watt-hours.