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Hi, we can help with that. Do you know how many Volts and Watts that you want the kit's motor to be?

If you would like help choosing the right motor size motor then please let us know how much weight the cart will need to carry and we can then run some calculations to determine the optimal motor size.

I found a kit that looks like it is very close to being what you need with the exception of its freewheel and freewheel adapter. It is item # KIT-328-C and here is a link to it: 

https://cart.electricscooterparts.com/24-volt-350-watt-electric-beach-wagon-power-kit-KIT-328-C

We have 3/4" axle adapters that a sprocket bolts directly onto so the cart can have reverse and I can make a new version of the kit above with a reversing controller and the sprocket and axle adapter that you need. 

So I can narrow down which sprocket and adapter are needed for the kit have you run the calculations to determine what size axle sprocket you need?

If you would like help with these calculations then please let me know the outside diameter of the motor-driven tire and I can help with that. 

I ran a top speed and gear ratio calculation with the 350 Watt gear motor and found that a 44 tooth axle sprocket would provide a 3.59 MPH top speed which is an ideal top speed for a beach cart. Here is the calculation.

We have a 44 tooth sprocket and non-freewheel adapter for 3/4" axle so that part of the kit is available.

The two 12V 15Ah batteries in KIT-328-C are estimated to provide a 35-minute runtime under extremely demanding condition such as continuous deep sand and hills. If the cart is used under less demanding conditions part of the time then the runtime is estimated to be longer than 35 minutes. For example on flat ground and hard packed dirt or pavement the estimated runtime is 75 minutes. I can make a new version of the kit with larger 22ah or 35Ah batteries for a longer runtime, or with smaller 10Ah batteries for a shorter runtime. 

The on/off power switch and the reverse switch can be on the handlebars, the kit could also have a key switch which is built into the throttle if you prefer that to a push button power switch. 

Please let me know what your preference is on the battery size, and if you would like the kit to have a push button on/off power switch or a key switch on/off power switch, and then I can get started on making the page for the kit. If you have any other requests for the kit then please let me know also. 

Please let me know if you have any questions. 

I forgot to mention in my previous reply that the hauling capacity for a 350 Watt motor geared for a top speed of 3.5 MPH is estimated to be around 250 pounds under the conditions which you mentioned.

With a beach cart that has reverse and no freewheel, the motor will provide a significant amount of rolling resistance when the cart is pushed or pulled by hand without the motor running. The motor could be mounted on a hinged plate that is held down by a thumbscrew so the chain could be easily removed in the event that the batteries go dead.

With a freewheel in the drivetrain, the cart can be pushed forward without any resistance from the motor and when the cart is pushed backward the freewheel will engage and turn the motor. This is just like a bicycle with a freewheel where the pedals do not move when rolling the bike forward, however, the pedals do move when rolling the bike backward. 

We can make the kit with either reverse and no freewheel, or with a freewheel and no reverse. 

Here is a link to the kit with 3/4" axle adapter: 

https://cart.electricscooterparts.com/24-volt-350-watt-electric-beach-wagon-power-kit-KIT-328-C-1