Yes, it is possible to create electromagnetic waves using magnets. No, it is not possible to create magnetic waves without an electric field being present. Electric fields are created by electric charges. For instance, if you statically charged up a balloon by rubbing it on your hair, the balloon creates an electric field. Magnetic fields are created by magnets. For instance, a fridge magnet creates a magnetic field and uses it to stick to your fridge. Electric fields and magnetic fields are not separate entities. They are really facets of one unified entity: the electromagnetic field.