No, because,in the absence of the interactions with the Higgs, it's not possible to give a mass to the leptons, since the left handed leptons have different properties than the right handed leptons-that's why the Higgs is necessary in the first place.

If one neglects weak interactions completely, then, in QED, in which left handed and right handed components of the electron transform in the same way, electromagnetic interactions cannot generate a mass term, due to the global chiral symmetry. In principle this symmetry could be broken and a ``theta'' term generated (that corresponds to a  E.B term), that could, then, contribute to a mass term for the leptons. However the coefficient of this term is very strongly constrained by the fact that it would contribute to an electric dipole moment for the electron, that has been measured to be very small, compatible with zero. Therefore, the final answer is No. 

Global chiral symmetry is broken. Because if it is present it would forbid the decay of a neutral pion to two photons. However experimentally we observe that Γ(π0→ 2 γ) ≈ 8.4 eV. On the other hand the electric dipole moment of electron is d=(-0.1 ± 0.9) x 10-19 e-cm. Perhaps this smallness of d is due to the presence of yet another symmetry which controls the strength of the θ term in QED which should be small but non-vanishing.

Chiral symmetry breaking by ABJ (Adler-Bell-Jackew) anomaly introduces a new mass scale in QED. But this has to be much smaller than observed electron mass. Therefore one can safely disregard electromagnetic self mass of an electron.

Once more: while the global chiral symmetry of QED, were the electron massless, could be, in principle, broken and a theta term, thereby, generated, which would, then, lead to a mass term for the electron,  it's an experimental fact that it isn't-that the coefficient of the theta term, in QED, for leptons,   is consistent with zero. So chiral symmetry breaking, in fact, doesn't occur in QED in the lepton sector-though it could. 

What would be the mechanism in QED that would solve its CP problem (which this would be) is, indeed, rendered irrelevant by recognizing that the correct framework for describing electromagnetic interactions is the electroweak theory, in which the way for describing the mass of the electron is completely different. Within that theory, the coupling of quarks and leptons to the photon is realized in a very specific way. 

We should not view QED as an isolated theory and rather discuss electromagneitic interaction as a part of a electroweak theory. This theory is SU(2)L x U(1)Y invariant and below the energy scale of electroweak breaking (Mw or Mz) QED becomes a correct theory. That is Mz or Mw becomes a natural cut-off scale for QED to be valid.

In this context Higgs mechanism is the usual way to break SU(2)L x U(1)Y symmetry and that way electron also gets it's mass of  half MeV or so. Yukawa coupling which is required for this to happen is, unfortunately, again very tiny (10-6 or so).

The HIggs boson seems not to be required at all to determine the mass of electrons and positrons, since it has been known since the 1930's that any photon of energy 1.022 MeV or more, that has no rest mass and is electrically neutral, will destabilize and convert to a pair of electron-positron, massive and charged in opposition, when grazing a heavy particle such as an atomic nucleus. 

Confirmed by Blackett and Occhialini from analyzing recorded cosmic radiation scattering impact traces in a bubble chamber in the early 1930’s, and extensively confirmed ever since.

Any energy in excess of the 1.022 MeV converting to the two 0.511 MeV/c^2 massive particles is related to the velocity in opposite directions of both escaping particles.