Question

How come an atom cannot have a permanent dipole moment?

Answer 1

It is because the distribution of the electrons in it can arrange itself in such a way to become symmetric. Then atom's components do that: they assume a symmetric shape, minimizing the coulombian potential. A spherically symmetric distribution of charges has not a dipole.

Explanation:

The minimization of coulombian (electrostatic) potential, leading to symmetric distribution of charges, can be established for a classic distribution of charges using Coulomb's law.

Analogously, for every quantum wave function - s, p, d, f type, it can be demonstrated that they are centrosymmetric; therefore apolar.

Then, any combination or "sum" of electron distributions of such centrosymmetric wave functions give rise to an unpolar centrosymmetric distribution of charge.

Moreover, it can be demonstrated that the sum of all the wave functions of the same quantum number `l`, with all the different quantum numbers `m_l`, corresponds to a spherically symmetric distribution. For example, if you sum up the three wave functions of `np_x, np_y, np_z`, having `l=0` and `m_l = -1, m_l = 0` and `m_l = +1` respectively, give rise to a spherically symmetric electron cloud, similar to the one of a `ns` orbital, for any `n`. That is the probability density depends on the radial distance from the nucleus only, and not on the angle or direction.

Your interesting question and related answer has an important follow-up on the ability of an atom to emit light quanta.
Analogously to the classical case, in which only an oscillating electric dipole can emit electromagnetic waves, the excited atom would emit a quantum of light only in the "transient" switching of an electron from a kind of centrosymmetric orbital to a different kind of centrosymmetric orbital. We can represent to ourselves that transition as the only instants in the "life of an atom" in which it becomes asymmetric (and polar) .
For example, the electron transition from a `3p_x` to a `2p_x` orbital wouldn't cause that transient dipole. This means that the atom can't get rid of the decrease in energy by throwing it away as a quantum of light. Therefore that transition simply does not happen (it is a "forbidden transition"). This is stated in a quantum selection rule: `Deltal = +-1`.