Electromagnet for string vibration
You do not need permanent magnet here indeed(well, in theory).

Though using only coil can be problematic since magnetic field from it is very modest, you would need to place the coil very close to the string to vibrate it and it is not practical.

Permanent magnet helps simplify the coil, by changing the magnetic field generated by coil you would change the magnetic field of permanent magnet, therefore vibrating the string.
With this setup you can make coil suitable for amplifier(8 ohms, ~1 Henry for example) to work with.