I would not store a cart with a hoist system. Wooden vehicles are not designed to take stresses of weight, being pulled from above. That constant downward pull is really hard on any kind of wood, it has to give. Carriages are built to take very different kinds of stress and strain, while in motion on the roads. Wood parts will give or torque, and the shape will end up twisting in the body. Will happen slow or fast, depending on weight and humidity in the air.
Shafts are recommended being stored hanging down or laid flat for 4-wheelers. 2-wheeler vehicle carts need LEVEL support under shafts while in storage. Shafts might also need a strap around them, to hold the bent areas apart the correct distances. Shafts with sharp curves or hooked ends are usually seen with metal strap support underneath the bent area, to hold them in that shape. They still need LEVEL support out in front of the cart, to prevent one high or low shaft while in storage.
For the vehicle shown by Yip, I would put a sawhorse under those shafts to level them up, get the weight off the tips on the ground. You could also tie the shafts to the ceiling, probably midway (thicker area) down the shaft length. Then making sure they are the same height from ground, level, when you tie them off. Tying them up by the tips to the ceiling, is the same as putting tips on the ground, weight is pushing on the thin, very flexible end parts.
Not much more pitiful than a cart or buggy with shafts that tip up at the ends, from being stored with shaft tips on the ground.
Wood is extremely flexible, can change very fast in poor storage. Wood absorbs and looses water according to the air around it, all an on-going process. Wood exposed to ammonia smells, by storing in the stall area, will make the finish, paint or varnish deteriorate faster than expected. May make other things happen to the wood also, like rot. Might be a reason to look at the newer vehicles, with metal wheels and bodies. They can also have paint problems, some rust, in ammonia exposure storages.
Quite a few folks do store their one vehicle in the trailer, but trailer is cleaned out, dry floored, NO BEDDING. If you have little or no other storage space, you have to live with the problems that come up. Maybe a rental unit to put the wood cart in for long times of non-use, like snowy winters.