Why can't we have charge field? I learnt that everything is made of field, there is the electron field that occupy the entire universe and it will produce an electron which is an excitation of the field. This electron interacts with the higgs field that give it it's mass and it also carries a negative electric charge, I am wondering why can't an electron through interaction with a say charge field instead? I also know there is the electric field but you need to have a charge to create an electric field just like a mass creating gravitational field. 
 A: Charge and mass are very different properties because charge originates from a fundamental symmetry while mass does not. Indeed a non-zero mass breaks a fundamental symmetry. This difference is why we believe the two properties have different origins.
Charge arises from a symmetry of the Dirac Lagrangian This is nicely described by JamalS's answer to What is charge actually? How to define it? In the Standard model it is associated with the $U(1)$ symmetry. Any theory with this symmetry will inevitably have a property homologous to electric charge. No external mechanism is needed to produce it.
By contrast mass is not a fundamental property and is not required by any symmetries. Indeed the $SU(2)$ symmetry of the Standard Model is only possible when all particles are massless. So we believe that all particles are fundamentally massless and mass only exists because there is some external mechanism to produce it. That of course being the Higgs mechanism.
So at least in quantum field theory the answer to your question is that no, charge is not produced by a mechanism akin to the Higgs.
A: We do have charged fields, such as the electron, which you mention. Do you mean can the electromagnetic force be attributed to a type of field called a charge field? Well it sort of can (at a stretch), it's called the photon field. 
The photon field appears in quantum electrodynamics or QED, which is our quantum field theoretic description of the electromagnetic force. The photon mediates the electromagnetic force between charged fields. 
I must stress though that the photon doesn't interact with itself. That means that it isn't itself charged. So calling it the charge field could be quite misleading. 
The photon field, however, shouldn't be thought of as the ''origin of charge'', in the same way that the Higgs field could be said to provide the ''origin of mass.'' The mass and charge assignments of the known particles are both puzzles; we don't know why they have the values they do. However, whereas the existence of mass itself is a problem (solved by the Higgs), the existence of charge isn't a problem. Thus there is no reason to expect that any field plays the role of the Higgs with respect to charge.
A: Charge is not a field; charge is a coupling constant measuring the probability of interaction between two fields: a matter field and a gauge field.
For example, the electric charge of an electron, $-e$, measures how strongly the electron field couples to the photon field... basically, how likely an electron is emit or absorb a photon.
Amother way to think about it is that fields correspond to particles, the particle being a quantum of the field. There is no “electric charge particle”, so there is no “electric charge field”. Instead, various particles have various electric charges depending on how strongly they couple to photons.
