Electrons (and, their cousins Muon and Tau) carry Weak Charge having value $-1/2$.
If you believe in Strong Anthrophic Principle
Why does electrons carry Weak Charge?
If you don't believe in Strong Anthrophic Principle
Does electrons participate in Weak Force interactions naturally? Is there a mechanism known with which electrons get Weak Charge?
There are a few different things which you might refer to as "weak charge," and I'm not sure which you've chosen here. The electron and the other charged leptons are members of weak isospin doublets; they must have total weak isospin $\frac12$ to be able to rotate into their corresponding neutrinos, an interaction mediated by the $W^\pm$ boson.
Wikipedia thinks "weak charge" is weak hypercharge, which is $-1$ for left-handed leptons (charged and neutrino both) and $+1$ for left-handed baryons; the hypercharges for the right-handed particles are different to account for the fact that a right-handed neutrino would have zero weak hypercharge and not participate in charged-current weak interactions.
I have a professional bias for considering the "weak charge" as something analogous to the electric charge: a parameter intrinsic to a particle which determines its coupling to the weak neutral current. In this scheme the electroweak charges are $$ \begin{array}{ccll} \text{particle} & \text{electric charge} & \text{weak charge} \\ e & -1 & -1 + 4 s_W^2 &\approx -0.041 \\ \nu & 0 & +1 \\ u & +\tfrac23 & +1-\tfrac83 s_W^2 &\approx +\tfrac13 \\ d & -\tfrac13 & -1 +\tfrac43 s_W^2 &\approx -\tfrac23 \\ \text{proton}=uud & +1 & +1-4s_W^2 &\approx -0.041 \\ \text{neutron}=udd & 0 & -1 \end{array} $$ Here $s_W^2 = \sin^2\theta_W \approx \frac14$ is related to the weak mixing angle, a fundamental parameter of the standard model whose value depends on momentum. Note that for the electron and proton the weak charge is almost zero; in fact the difference between the low-momentum $\sin^2\theta_W = 0.2397$ and the high-momentum ("$Z$-pole") $\sin^2\theta_W = 0.23120$ changes the weak charge for $e$ and $p$ by about a factor of two.
These charges arise from the coupling terms in the Standard Model Lagrangian.
As mentioned by Jerry Schirmer, electrons are involved in the beta decay of nuclei. This reaction goes as follows:
$$ n \rightarrow p + e^- + \bar{\nu}_e $$
The modern theory of (electro)weak interactions treats them as Yang-Mills interactions based on the $SU(2)\times U(1)$ gauge group. In the reaction above a new intermediate force carrier, the $SU(2)$ gauge boson (specifically, $W^-$) is introduced:
$$ n \rightarrow p + W^- \rightarrow p + e^- + \bar{\nu}_e $$
In order to couple neutrinos and electrons to the weak bosons, we must specify their weak charge. The structure of the gauge theory explains it naturally.
It is also worth mentioning that only left-handed electrons have non-vanishing weak charge. Right-handed electrons, on the other hand, don't interact with weak bosons. It can be seen even from the equation above, since chirality is conserved and anti-neutrinos are always right-handed.
