How can moving electrons participate in electrostatic interaction? People say that there is an electrostatic force between electrons and atomic nuclei. However, electrostatic force applies to static charges, i.e. charges at rest.
Question:  How can electrostatic force exist given that electrons aren't stationary?
 A: The Coulomb interaction remains valid when velocities are much smaller than the speed of light.  This is called the "quasistatic" regime, and it's the one that applies in the analysis of the electron-nucleus interaction.
A: When talking of electron and nucleus, one has to use quantum mechanics to model the interaction. In quantum mechanics the coulomb potential enters the differential equations which will give the eigenfunctions of the system . 
The Bohr model was developed on the classical thoughts, as described by Zero , but the correct model is the quantum mechanical which gives the spectra of the atoms and the probabilities of interactions.
A: To a good approximation the nucleus can be considered at rest and the nuclear potential to be static.
A: You are absolutely correct, it does not make any sense to use the electrostatic force on an electron racing around the nucleus.
The electrostatic force was formulated by Coulumb and is therefore also called Coulumb's force. Coulumb measured the attractive and repulsive forces between charged iron spheres. In such a charged object, electrons will repel each other, and the force work in all directions. In a stationary charged iron sphere, force, current and magnetic field, goes in all directions, as the individual electrons move in all directions.
For a current in a wire we can't use the electrostatic force, as the charges are moving in the same direction. As the current move in one direction, force 
and magnetism act in perpendicular directions as described by the right hand rule. For charges moving in the same direction we therefore use a simpler version of the electromagnetic force.
It was Niels Bohr which used the electrostatic force as the force between the electron and the nucleus. And with it he constructed a solar system like Newtonian model of the atom, where electrons are confined to Bohr orbitals. But now we photograph the atom, and there are no Bohr orbitals, we observe an electron haze, so Bohr's assumption was wrong.
So we might update Bohr's postulates with our new observations. If we simply multiply Bohr's electrostatic force with the Bohr radius, we set the electron free from the Bohr orbitals and allow electrons to orbit the nucleus in an electron cloud:

We can then also see how the electromagnetic force works different in an atom, in a current and in a charged object, all due to the elementary charges, adding up. The relationship on how elementary charges add up can be described like this:

By using this simpler electromagnetic force Quantum Mechanics can be solved with simplified orbital mechanics. It even gives a new and simple term of the fine structure constant. Here is a link to the publication which solves QM in the simplest way: http://vixra.org/abs/1804.0050
