Although the electron may not be theoretically massless in theory, I believe I understand what inspired this question.
All fermions, both quarks or leptons, start in the Standard Model as massless fields. They do not possess an intrinsic rest mass, which is to say that they don’t have one. Particle physicists would also tell you that their Lagrangians are massless. However, charged fermions can interact with another field. This complex-valued scalar triplet is called the Higgs field.
It is important to note that fermions have massless properties. This makes it possible for the theory of relativity to be “renormalizable”. This means that a naive calculation using theory can result in inconsistencies, but they can be eliminated by a mathematically rigorous, systematic process. If the fermions in the theory had mass, this would be impossible.
Many interesting properties are found in the Higgs field. It is responsible for electroweak symmetry breaking. When they are not subject to excitations (particles), most fields are at their lowest energy level. The electromagnetic field, for instance, is at its lowest energy level when there are no photons. The Higgs field is not in this category: Its lowest energy status is when there are excitations. The vacuum can also decay, or enter a lower energy state, by producing excitations from the Higgs field.
This is the new “vacuum”, which has the lowest energy state. This “vacuum” is then populated by charged fermions. (The particle physicist explains that the Higgs field within this vacuum has a nonzero vacuum expectation value, or v.e.v. This interaction behaves as if the charged fermions (including the electron) had mass.
The theory doesn’t say that the electron has no mass. It does however state that the electron’s mass is not an intrinsic rest mass, but an effective rest mass, which results from the electron’s interaction and the Higgs field, and specifically its nonzero vacuum expect value.