Raising funds for this charity is a worthy cause but be careful that you don't harm cycling in the process. Cycling does not have to be such an unpleasant struggle that people have to give you money to compensate for your pain. Cycling is enjoyable and good for you and worth doing with or without a charity cause.
Pick a bike that is suitable for the training as well as the ride. A winter training or light touring style roadbike is good. Make sure that it fits you, has sufficient low gears ( not just quantity of gears) and has tyre clearance for mudguards + wide enough tyres. I would suggest 28mm as good, permitting narrower ones if you want.
Start your training short and easy and gradually ramp up the distance or time over weeks. Any time in the saddle is good. Fit some pannier racks and luggage so you can commute or shop by bike.
Essential kit is probably helmet, shorts, gloves. In winter, non-bike wicking t shirts and light fleece are fine. Get some bright shell layers, one waterproof and one windproof. Ronhills, Aldi Go Outdoors all do cheap durable bike clothes.
If you are looking for a loan bike from someone throw a bone, post your height.