A frog has already been levitated without any magnetic object attached to it. I'm wondering if it's possible to reverse this for transportation purposes, by having a vehicle levitate itself using magnetism, without the usual magnetic track. I.e., is it theoretically reasonable to make some sort of vehicle (car-sized or larger) produce a strong enough magnetic field to levitate and move itself over normal ground?
The Frog Levitation works because water is Diamagnetic (It weakly repels and is repelled by magnets) and frogs are mostly water. Other materials may be Paramagnetic (weakly attract/attracted by magnets), Ferromagnetic (Strongly attract/attracted by magnets, and able to become permanent magnets) or Superdiamagnetic (Repel magnets fairly strongly, superconductors do this)
The magnet needed to levitate the water in a frog is much, much heavier than the frog. A magnet is not going to be able to levitate itself over a diamagnetic surface like water, and even if you had a stupendously powerful magnet that could do this, it would only work over diamagnetic surfaces. The same magnet would pull on paramagnetic materials with comparable strength and would have an immense pull on ferromagnetic materials, including all iron and iron alloys.
A hypothetical magnet able to exert it's own weight of repulsion against water at an interesting distance is going to be enormously dangerous at a much greater distance for any form of iron. Its movement (Which is rather the point of a vehicle) will also induce electrical currents in any conductive material around it which would produce a sort of electromagnetic drag. My understanding is that the more powerful the magnet, the faster it moves, and the more conductive the conductor, the worse this effect would be. This is not my area of expertise though but I expect that if a permanent magnet dropped down a copper pipe shows a highly noticeable effect, then the stupendously powerful electromagnet needed for diamagnetic levitation would suffer from this quite severely.
is it theoretically reasonable to make some sort of vehicle (car-sized or larger) produce a strong enough magnetic field to levitate and move itself over normal ground?
Possible? Maybe. Reasonable? No.
I'll let someone better acquainted with the physics discuss what you would need to generate the necessary field to accomplish the feat.
But practially it's not going to happen compared to the current technology. The reason that the tracks will continue to exist is because the equipment to generate the power to drive the field, the coolant to keep the magnets happy, the equipment to control the field to drive the train, the magnets themselves... they're all damned heavy. By putting all that heavy stuff on the thing you're trying to lift is making it that much more difficult (read: costly) to lift it.
You are essentially asking if it is possible to create a Hoverboard, and the answer is yes. Magnetic suspension works by.....magnets. One magnet repels another.
Theoretically, if the surface you intend to 'hover' over is a ferrous (containing iron) material with at least a slight magnetic field, you can use an electromagnet in the hovering vehicle to create a field strong enough to propel that vehicle into the air.
This would would require a very lightweight and powerful electromagnet to be attached to the hovering vehicle, and is not feasible with today's technology.
Also, the hovering vehicle would stop once the magnetic field of your base object (or the Earth if hovering over ground) dissipated or weakened enough.
And rememebr, hoverboards don't work on water....