Electric field is determined by the electric force per unit charge, but photon clearly doesn't carry charge so where do the electric field came from?
Photons don't generate electric field. Photons are electric field, or more precisely electromagnetic field.
In classical physics we tend to describe quantities in terms of fields, which are entities that span over the whole spatial dimensions and contain information at each point in space. You gave the example of the electric field, which contains the information about the electric force on sample charge at each point.
In quantum physics, or more specifically in Quantum Field Theory, we usually describe quantum fields as collections of particles. The procedure that takes a classical field and tells us how to deacribe it in terms of field particles is known as Second Quantization. In the case of electromagnetism the theory is called Quantum Electrodynamics (or QED for short). It tells us that the electromagnetic field itself is composed, in fact, of excitations of particles named photons. In other words, a photon is an excitation of a specific state of the electromagnetic field.
It is not necessary to have a charge in order to generate an electric field. One relevant Maxwell's equation includes the curl of E related to dB/dt, and another adds a term to electric current called the 'displacement current' that generates a magnetic field (B) from a nonconstant electric field (D), even if there is no current of charges in motion.
So, a photon is created in a light bulb by charges (electrons) in thermal motion, but once it starts propogating it leaves that behind. The photon needs carry no charge in order to travel (indeed, it travels faster in a vacuum than any charged particle).
Historically, the notion that some material supported the wave motion was rejected by Einstein on philosophical grounds, and found inconsistent with results of the Michaelson-Morley experiments of 1887. Those were very important steps in early 'modern physics'.