Would it be possible to create a constant EM field that disrupts electronics within it in a similar manner to the way an EMP from a nuclear blast does?
As several comments have pointed out: yes, a constant EMP would be impossible. However, a device with consistent EMP-like effects, is certainly possible.
An EMP is an ElectroMagnetic Pulse. This works due to the fact that a change in current, causes a change in magnetic flux, and a change in magnetic flux causes change in current (Faraday's law).
An EMP works by creating a powerful magnetic pulse (fluctuation), causing a change in current in an electronic device. Now, if you wanted to make this consistent, you would merely have to constantly fluctuate the magnetic field, causing a constantly fluctuating change in current.
Or would it be possible to have a planet where the magnetic field of the planet is strong enough to prevent electronics from working on all of the planet or in certain areas of the planet?
Unfortunately, a planet's magnetic field is (fairly) constant. Because of this a large amount of magnetism coming from a planet wouldn't stop electronics from working. Despite this, @mviereck pointed out in the comments, a pulsar is a natural phenomenon that creates an immense amount of magnetic field, while rotating around its axis. This is, basically, a giant EMP.
However, as @Alexander pointed out, either a large pulsating EMP, or a pulsar could be easily overcome by using a faraday cage.