We have a theory of a Higgs field that describes how a particle gets mass. Since mass and charge both are intrinsic properties of a particle, is there any similar theory for how particles get electric charge?
Your question is very good. The answer is very easy, but very deep though.
"Charge $Q$" means that there is a conserved (quantum or classical) number $Q$ respect to some symmetry $G$. i.e. the system is invariant respect to certain symmetry $G$ transformation. (You can derive it from Noether theorem or simply the transformation on the "fields" in general, providing there is no quantum anomaly due to path integral Jacobian varies or the anomalous currents.)
The symmetry can be gauge (local) symmetry or global symmetry. The gauge can be the normal small gauge or the large gauge transformation. The large gauge transformation is very similar to the type of global symmetry transformation.
1. In E & M, we have U(1) gauge symmetry. We thus have electric charge $e$, $-e$.
2. In QCD, we have SU(3) gauge symmetry. We thus have three color charge $r$, $g$, $b$ and its anti-colors.
3. In quantum gravity, some people had believed there is no global symmetry, everything is local gauge symmetry.
Although theories of electromagnetism (a la Maxwell) and electroweak unification (a la Weinberg, others) work well to explain the behavior of charge / magnetism / electroweak, none of these provide a mechanism for originating electric charge. Similarly, in QCD, which explains the behavior of the strong force between quarks and gluons, no mechanism is described for originating "color charge".
Although the Higgs mechanism certainly gives mass to quarks, electrons, and the W and Z bosons of the electroweak force as the last missing verification of the Standard Model, it does not explain the majority of mass / energy of bulk matter. Most of that derives of virtual particle interactions that occur inside of hadrons and mesons, and for the strong force, something like the Higgs field or mechanism is evidently not required. Neither does the Higgs field or mechanism yet provide us with any deeper insight into the origin of gravitational mass or the riddle of dark matter or dark energy, even though in classical physics, inertial mass and gravitational mass were believed to be intimately related.