Whilst there are not any specific detailed plans you can download and build your kart to, there are quite a few guidelines and sketches that can be used as a starting point. Ideally all karts are a unique expression of the owner, each individually crafted to mimic or be inspired from an inspiration car dating from the turn of the last century. However not everyone has an automotive manufacturing background or paid attention back in metalworking class at school, and so a basic generic chassis and drivetrain design that you can base your creation on is a good idea.
The classical CycleKart design is known as the 'Stevenson' design. This name is attributed to the guy(s) that penned it and is the easiest style of kart to build. It has straight chassis rails, welded together with a slight inward taper from the rear to the front. The passenger compartment is generally a plywood box that sits between the rails. The bodywork is an extension of this basic box in whatever style the owner is trying to achieve.
The classic 'Stevenson' Cyclekart design
A CycleKart built to the Stevenson design using the MG P-Type as the tribute car
There are also many variations on this basic design with profiled chassis and more ornate bodywork but the basic wheelbase and track of the kart remains essentially the same. There is no real limit to what you can do as long as your CycleKart follows the basic specifications as detailed in the CycleKart Specifications
Most people that are building a Cyclekart to the Stevenson design, overlay an image of their tribute car on top of the basic design blueprint above, so that they can scale the shape and dimensions of the body. You can find a step-by-step process of how to do this in the article "How to overlay an image of your tribute car"
You can find some additional plans and discussions on chassis design in the forums HERE
Generally CycleKarts are representation of actual cars from the period. However, CycleKarts that are not based on specific cars, but are constructed in the style of cars of the period are also acceptable.