Can one assign an equivalence principle of some kind to the EM field?

The fact that gravitational field can be simulated/canceled by inertial forces relies upon the following elementary but fundamental fact.

The gravitational coupling constant of a given body, i.e. its gravitational mass,$M$, coincides with the other universal constant associated with that body, appearing in the general law of motion, i.e. the inertial mass $m$. So if a gravitational field $\vec{g}(t,x)$ is given, the equation of motion of a body with mass $m$, immersed in that field of acceleration is, $$m\frac{d^2\vec{x}}{dt^2} = M\vec{g}(t,\vec{x})$$ and, since $m=M$ $$\frac{d^2\vec{x}}{dt^2} = \vec{g}(t,\vec{x})\:.$$ The motion, therefore, depends only on initial position and velocity of the body but not on other properties. Exactly as geodesics do in a spacetime. So, a description in terms of geodesics in spacetime is allowed this way and the geometrization of gravitational theory enters physics.

Referring to Electromagnetic field, this story stops at the first step. Indeed the corresponding of gravitational mass is the electric charge $q$ and, evidently, $q \neq m$ and so, $$m\frac{d^2\vec{x}}{dt^2} = q\vec{E}(t,\vec{x})\:,$$ whose solution depends on the ratio $q/m$, not only on the initial position and velocity.

This is the reason why there is no equivalent of equivalence principle for electric forces and any attempt to geometrically describe electromagnetic interaction must be constructed following other approaches (gauge theories) without involving things like metrics and geodesics.

To give a short answer: There is a huge geometric framework behind electromagnetism. This framework is gauge theory!

The leading idea is that you have electromagnetism as the gauge theory of a $U(1)$ Lie group. To keep the theory invariant under local $U(1)$ transformations, you introduce a connection (the gauge field $A_\mu$ which is identical to the four-potential) and a gauge curvature.

This curvature is exactly the $F_{\mu\nu}$ in QED. This corresponds to the Riemann tensor.

Another way to look at it is to see gravity as the gauge theory of the Lorentz group, but that would lead off-topic here.

It is possible to treat electromagnetism in a directly geometric way. This theory is known as Kaluza-Klein theory and it works by applying Einstein field equations in a 5 dimensional spacetime, and then taking the 5th dimension to be 'small' in some sense.

A nice review can be found here:

Christopher F. Chyba: "Kaluza–Klein unified field theory and apparent four‐dimensional space‐time", Am. J. Phys. 53, 863 (1985)

Of course, as the paper states: "Whether the theory represents more than an elegant curiosity remains unclear...".