Assuming the fields die off asymptotically at infinity, you can write the action for a charged point particle with coordinates $X^\mu(\tau)$ (where $\tau$ is the particle's worldline) as
$$
S_q = q \int d^4 x \int d\tau \delta^{(4)}(x^\mu - X^\mu(\tau)) A_\mu(x) \frac{d X^\mu}{d\tau} = q \int d\tau A_\mu(X) \frac{d X^\mu}{d\tau}
$$
You can add this to the action for a free point particle
$$
S_{pp} = m \int d\tau = m \int d \tau \sqrt{-\eta_{\mu\nu} \frac{d X^\mu }{d \tau} \frac{d X^\nu}{d\tau}}
$$
and the Maxwell action for the fields
$$
S_{EM} = \int d^4 x \left(-\frac{1}{4} F_{\mu\nu} F^{\mu\nu}\right)
$$
to derive an action with an electromagnetic field coupled to a point particle
$$
S\left[A_\mu(x), X^\mu(\tau)\right] = S_{EM}[A] + S_{pp}[X] + S_q[A, X]
$$
Following some interesting discussion in the comments, I should mention that this action is meant to be understood in an effective field theory sense, valid when the relevant wavelengths/physical scales are larger than the true size of the charged particle. Classical physics tends to become singular and break down near true point particles, requiring some kind of regularization; this will happen if "self force" effects become important. In other words, the above action (like the Lorentz force law for a point particle) implicitly assumes you are going to be looking at the motion of a point particle in some external field, and not worry about the effect that the particle's field has on itself. One avenue toward dealing with finite size effects is to include higher order corrections in the effective theory, for example as described in Section 2.6 of arxiv.org/abs/hep-ph/0701129 in the context of gravitational waves. However, ultimately if you want to probe length scales that are the size of the particle, the point particle approximation is bound to break down. A resolution to this within classical mechanics would be to say the point particle actually has some finite radius. However, when dealing with, say, an electron, the actual resolution is that quantum field theory becomes relevant at some scale. The "self energy" of the electron can then understood in terms of Feynman diagrams with loops, that renormalize the mass of the electron. Anyway, the point is that the above action is to be understood as an approximation that is valid so long as you are willing to consider physics coarse grained over large enough length scales (just as with the Lorentz force law, which the above action reproduces).