Ah, the electric force is present in those effects, but in a more local form. I believe understanding that alternate form is the solution to your question.
The electric force is often first encountered as an inverse-square law between distant charges. This is perhaps the description you know: two charges instantaneously experience repulsion or attraction when separated by some distance.
However, this "action at a distance" was theoretically unappealing to many physicists, who desired a local description of nature. Sure enough, physicists found a way to expand and reformulate E&M as a theory about electric and magnetic fields: because fields spread throughout space, a charge will generate a field which propagates outward and then interacts locally with other charges (and vice-versa), all without any action at a distance! This formalism also let's us fuse electric and magnetic forces into a single electromagnetic force, and in doing so reveals the existence of electromagnetic waves. These waves were identified as superpositions of photons, the fundamental constituent of light. In fact, any electromagnetic field may be reinterpreted as a swarm of (mostly 'virtual') photons.
All that is to say: charges do not interact directly with each other, but via the exchange of photons. Charges absorb and emit photons, and that's how they experience the electromagnetic force. The same interaction that admits this pushing and pulling is the same interaction that appears in the photoelectric and Compton effects.