Is this feasible?

westend said it all. Buy a used RV and do some repairs/modifications.

Using a lawn shed as a RV will give you problems.

However, it is an interesting thought and, I give you credit for being innovative and thrifty.
I don't think it's practical.

This is one of those ideas that sound like it may save a bunch of money but will be less than optimal, if proceeded on.

First, you lose the use of the shed on your property unless you're willing to unload it every time you're home or will have to climb up into it when using for storage. Both situations will have you looking somewhere else for storage.

Second, have you considered the weight of the shed? I used to build and sell yard sheds and maneuvering an 8 x 12 onto a flat bed and delivering put that endeavor out of business. I would guess, with conventional building materials, they weigh in excess of 5000 lbs. The four squares of roofing material weigh about 1000 lbs.

Third, shed versus RV trailer on the road, your shed may not like the high wind speed of travelling.

Fourth, do you plan to outfit this shed with any of the usual RV stuff like water, electrical, appliances, bed, cabinets, etc.? I think you'd find that putting in even half of that will make your costs greater than buying a used RV.

I'd suggest that you buy an RV and leave the shed on your property.