Unfortunately, the price of a plot of land close to Stockholm is usually high. And, even houses on wheel require building permits, if they stay permanently in the same place (or on the same plot), it does not help to move it on the plot, especially if there is no building on the plot.
As far as I remember, the time limit before a building permit is required, is six weeks. It also applies to caravans or other types of movable housing.
It is generally simpler to get a permit for any type of house on wheels, when there already exist a building for housing purposes on the plot.
So an advice would be to try to find someone who would rent you a place on their land, to place a house on wheels.
Or to find a long-term agreement with a campsite in the Stockholm area. It is quite common, to do this, for non-swedes that moves to Stockholm, and then start searching for some other type of housing.