Building your own horse cart is not particularly difficult--depending on your resources and skills. Horses have been used as vehicle motive power for carts and wagons and sleds for thousands of years in various cultures. The toughest part is figuring out just what you want to accomplish. Let's step through the planning process.
First step is to decide what sort of cart you wish to end up with. Do you want a Roman chariot to participate in historical reenactments or do you need a one horse racing sulky to show off your thoroughbred trotter? Let's design a light two wheeled cart for a single light horse and two passengers, using modern materials and construction techniques.
Do an internet search for "horse carts" and click on a few sites to find pictures and suppliers. The Amish people build and sell traditional carts and wagons. Others build pony carts, llama carts, dog carts, goat carts, etc. A few minutes spent here or in the library will give you lots of ideas.
The most difficult part will be obtaining the measurements, specifications, and dimensions for the cart you wish to build. Basically, you need to know how much weight you plan to carry and how big is your "engine". How tall is your horse? How long is he? How wide is he? There are "horsy" terms involved in all of this, like "hands", "withers", "flank" and "shoulder" and "brisket", etc. but you will need "inches" to come up with the dimensions for your cart.
Your cart will consist of a few basic parts. Lets start with the seat. I suggest a bench seat with a back. Make it out of three-quarter inch marine plywood and upholster it with foam and vinyl. It should be about four feet wide for two people. It needs to be pretty much weather proof. It should have metal bracing and mounting fixtures so that it can bolt to the frame. Take measurements from your favorite office chair to get an idea for seat width and back height. It should provide decent lumbar support if you are actually going to ride on it for long periods of time. There are no requirements for seat belts so far as I know--and I'm not sure you want to be strapped in anyway. Never saw a saddle with a seat belt, but who knows nowadays.
There will be two wheels. I suggest rubber-tired wire-spoke wheels, like motorcycle wheels. You could also use large utility wheels from the hardware store or light automotive wheels and front axles from the junk yard. The critical part is the strength of the axles and how they can be mounted to the frame. I suggest welding the axles in each end of a pipe or tube that can be clamped under the frame. For suspension we will depend on the rubber tires and padding on the seat. Additional shock absorption can be accomplished the old fashioned way with springs under the seat like the old fashioned buck board wagons or stage coaches. Putting springs between the axle and the frame introduces another degree of complexity for the home builder. The axles probably need to be at least three quarter inch diameter steel.
Now we come to the frame. I suggest one-by-two inch rectangular steel tube although round tube or angle iron will also work. The round tube looks very nice but it requires the ability to make fish-mouth welded joints. Angle iron must be heavier because it does not resist twisting as well as square or round tube. Aluminum alloy materials will make a very nice light frame but welding aluminum is problematic for the home builder. I suggest a simple square or rectangular frame with attachment points for the seat on top and the axle underneath. If either the seat or the axle can be adjusted frontwards or backwards, it will facilitate balancing the completed rig. Clamps probably work better than drilling holes through the frame. Holes will weaken it. You will need a place to rest your feet. Perforated metal or plywood mounted on the front portion of the frame will work fine.
Now we have a comfortable seat on a stiff frame on two easy-rolling wheels. How do we hook up to the horse? We need staves attached to each side of the frame and wide enough apart for the horse to fit in between comfortably. Smooth, round tube is nice for these. They must be long enough to extend beyond the horse's front shoulders. Your horse needs some room behind him. You sure don't want your cart running up and bumping him in the hind legs on a downhill section of road. That might cause all kinds of trouble. The staves should be parallel to the ground and about as high as your horse's belly. You don't want him stepping over them or bumping them with his legs. That will also cause trouble. Since your horse's belly is probably higher than your frame, you will need to figure out how you are going to mount the staves to the frame. The attachment point needs to be quite strong and rigid since all forces generated by the horse must transfer to the cart through these two fittings. A gusset-type fitting welded on the end of the staves and bolted to a similar fitting on the frame will allow for some adjustment in height for different size horses. A small dashboard structure between the staves on top of the frame can provide additional bracing as well as keeping your horse's hind feet from coming inside with you in an exuberant moment. Don't forget to use the geometry of triangles to provide strength at needed points.
The final part has to do with how your horse's harness and collar attach to the staves. You will need some eye bolts and/or rings welded to the stave to attach your harness fixtures. Your harness will use a collar and/or a breast strap for the horse to push against. There should be a butt strap of some sort so that he can hold the cart back in downhill situations. His harness will have some sort of back strap arrangement to hold everything level and in place.
Now you have the complete cart except that it is not very pretty. All the rough welded places must be smoothed up with a grinder and then it needs to be painted with a good enamel or lacquer suitable for outdoor conditions. Make sure there are no exposed sharp edges or pointy places that could hurt your horse.