One problem is, it doesn't seem that you can have a magnetic north pole without a south pole. You can get around this by having the magnet that is being orbited be a really big magnet, so that the other pole is very far away from the orbiting magnets. But you've still got the problem of the opposite pole of the smaller orbiting magnets. If the orbiting magnets rotate, the north pole of the orbiting magnet can get pointed toward the north pole of the magnet being orbited. I suspect this would make the orbits unstable.
So, actually, the electromagnetic force CAN cause orbits. Check out [this video](https://www.youtube.com/watch?v=UyRv8bNDvq4
), where they use static electricity (not magnets) on the Space Station (no gravity to interfere with the experiment).
It's just that, in general, regular matter produces gravity because the atoms are electromagnetically neutral (number of electrons = number of protons), so the electromagnetic force cancels itself out at the atom level.
Otherwise, EM is about [10^36 times stronger](https://en.wikipedia.org/wiki/Fundamental_interaction
) than gravity, so in the case of charged particles, the Earth's magnetic field [deflects the solar wind](https://en.wikipedia.org/wiki/Earth%27s_magnetic_field#/media/File:Magnetosphere_Levels.svg
) just fine, even though Earth has a gravity field too.