This answer uses classical general relativity and ignores quantum effects, so the electron is treated as a classical pointlike particle. Otherwise, we'd need to use a quantum theory of gravity, which is currently too difficult.
I suppose the charge of the black hole would build up much faster than its mass. Would such a black start repelling electrons at some point, because the electric repulsion would become stronger than gravity?
Yes. For convenience, we can use units of mass $M$ and charge $Q$ such that two objects with $M=Q$ neither attract nor repel each other overall. The charge-to-mass ratio of an electron is $q/m\sim 10^{21}\gg 1$. If we start with an uncharged black hole with mass $M_0$ and then somehow manage to add $N$ electrons' worth of mass and charge, the resulting mass and charge would be
$$
M=M_0+Nm
\hskip2cm
Q=Nq,
\tag{1}
$$
where $m$ and $q$ are the mass and charge of an electron. Now consider the interaction between this black hole and a single distant electron. Using Newtonian gravity, the ratio of electrostatic respulsion to gravitational attraction would be
$$
\frac{Q}{M}\times\frac{q}{m}
=\frac{Nq}{M_0+Nm}\times\frac{q}{m}
=\frac{x}{1+x}(q/m)^2
\tag{2}
$$
with $x\equiv Nm/M_0$. In this approximation, if $x\gg (m/q)^2\sim 10^{-42}$, then the electron will be strongly repelled because the ratio (2) is much greater than 1, even if we still have $x\ll 1$. This is true even when the electron is far enough away for the Newtonian-gravity approximation to be an excellent approximation, so this approximation is justified in hindsight. This approximation isn't good enough to tell us what the threshold value of $N$ would be, but it is good enough to confirm that electrons will be repelled if $N$ is large enough.
wouldn't it mean that the event horizon got smaller...?
I don't have a definitive answer. Here's what I do have. According to [1], the area of the event horizon of a charged (Reissner-Nordström) black hole with mass $M$ and charge $Q$ is
$$
A=4\pi R^2
\hskip2cm
R \equiv M+\sqrt{M^2-Q^2}.
\tag{3}
$$
If $Q>M$, then the event horizon is absent, exposing a naked singularity. According to equations (1) and (3), after adding $N$ electrons to an initially-uncharged black hole of mass $M_0$, the area of the resulting event horizon would be $4\pi R^2$ with
$$
R=M_0(1+x)\left(
1+\sqrt{1-\frac{x^2}{(1+x)^2}(q/m)^2}
\right)
\hskip2cm
x\equiv\frac{Nm}{M_0}.
\tag{4}
$$
This function $R(x)$ is a product of two factors, the factor $1+x$ being an increasing function of $N$ and the other factor being a decreasing function of $N$. The area initially increases with $N$, but beyond some threshold value of $N$, it begins to decrease. The question is whether or not this threshold is reached before further electrons start to be repelled. The analysis is not easy, for two reasons:
Newtonian gravity is not a good enough approximation in this case.
We can't use Hawking's area theorem, because that theorem assumes that naked singularities are absent [2], and the electron (treated as a classical pointlike particle) is a naked singularity because $q/m\gg 1$.
The closest analysis I found in the literature is the paper [3]. That paper considers a black hole that is already almost maximally charged ($Q\lesssim M$) and shows that we cannot make the horizon disappear ($Q>M$) by adding charged matter with $q/m>1$. Intuititively, this is because if $q/m$ is large enough to make the horizon disappear, then the matter will be repelled rather than attracted; and if we give it enough of a push (enough kinetic energy) to overcome this repulsion and force it into the black hole, then we have increased its energy enough so that $q/E$ is no longer large enough to make the horizon disappear. This doesn't quite answer the question, because the question is whether or not the horizon can be made smaller, and the paper [3] only explicitly asks whether or not we can make it disappear (and the answer is no). But the analysis in [3] is relatively detailed, and they also cite a couple of similar analyses, so with some effort, maybe those analyses can be adapted to answer the OP's question. The details are left as an exercise for the reader.
References:
[1] Section 8.6.5 in Straumann (2013), General Relativity (Second Edition)
[2] Page 6 in Düztas, "Cosmic censorship and the third law of black hole dynamics," https://arxiv.org/abs/1706.03927
[3] Sorce and Wald (2017), "Gedanken Experiments to Destroy a Black Hole II: Kerr-Newman Black Holes Cannot be Over-Charged or Over-Spun," https://arxiv.org/abs/1707.05862
