The electromagnetic interaction only drives the running of the electron mass $m(p^2)$, whereas the starting point of the running $m_0$ is determined by the Higgs mechanism.
Let's first consider the electromagnetic contribution (self energy $\Sigma(\not{p})$) to the electron propagator in the absence of Higgs field,
$$
G = \frac{i}{\not{p}- \Sigma(\not{p})+i\epsilon}.
$$
Without renormalization, the quantum loop corrections to the self energy $\Sigma(\not{p})$ would be infinite. However, with proper gauge fixing (Landau gauge) and regularization, it can be shown that the renormalized
$$
\Sigma(\not{p}=0) = 0
$$
as long as the fine-structure constant satisfies (Maskawa, Toshihide, and Hideo Nakajima, Progress of Theoretical Physics 52.4 (1974): 1326-1354.)
$$
\alpha = \frac{e^2}{\hbar c} < \alpha_{cr},
$$
where non-zero critical coupling is
$$
\alpha_{cr} = \frac{\pi}{3}.
$$
Since we know that electromagnetic interaction is "weak",
$$
\alpha = \frac{1}{137} < \frac{\pi}{3},
$$
it is not strong enough to induce spontaneous axial symmetry breaking (a non-zero $\Sigma(0) \neq 0$ breaks the axial symmetry). Therefore, electron remains massless. Note that the QCD color integration is "strong" enough to induce non-zero masses of quarks (not to be confused with the proton/neutron masses, which are resulted from quark confinement), even though it only contributes a small portion of quark masses compared with those from the Higgs mechanism.
Now enter the Higgs mechanism. The electron propagator in the presence of Higgs field reads
$$
G = \frac{i}{\not{p}-m_0 - \Sigma(\not{p})+i\epsilon},
$$
where electron mass
$$
m_0 = y\upsilon
$$
is dependent on Yukawa coupling $y$ and the non-zero vacuum expectation value $\upsilon = <0|h|0> \neq 0$ (which spontaneously breaks both the electroweak and the axial symmetries) of Higgs filed $h$.
Two comments on the self energy $\Sigma(\not{p})$:
- There are quantum corrections to $\Sigma(\not{p})$ stemming from the exchange of Higgs particles in addition to electromagnetic interaction. One can potentially regard the exchange of Higgs particles as the fifth interaction (depending on your definition of fundamental interaction). Nevertheless, this "fifth interaction" is negligible under most circumstances.
- In the same vein as the Higgsless case, the electromagnetic quantum correction to the renormalized self energy at zero momentum $\not{p} = 0$ is still zero $\Sigma(\not{p} = 0) = 0$ (I am not aware of any prove, anyone could point to any references?), given the "weakness" of the electromagnetic interaction. In other words, the mass $m_0 = y\upsilon $ is solely coming from the Higgs vacuum expectation value $\upsilon = <0|h|0>$. That being said, the physical mass $m_p$ (define by the pole of the electron propagator at mass shell) will receive a finite quantum correction arising from $\Sigma(\not{p}) - \Sigma(0) \neq 0$. This quantum contribution might be the "roughly 20% of the mass of the electron" Polchinski is referring to (mentioned in Mitchell Porter's answer as the renormalization group running of the mass). The electromagnetic interaction only drives the running of the electron mass $m(p^2)$, whereas the starting point of the running $m_0$ is determined by the Higgs mechanism.