$\newcommand{\ep}{\varepsilon}\newcommand\R{\mathbb R}$Let $L$ denote the set of all real random variables (r.v.'s) $X$ with log-concave density such that $EX=0$ and $EX^2=1$.
Let
\begin{equation*}
c:=\inf\{P(X\in(-1/2,0))\colon X\in L\}. \tag{10}\label{10}
\end{equation*}
We have to show that $c>0$.
For some sequence $(X_n)$ in $L$ we have $P(X_n\in(-1/2,0))\to c$ (as $n\to\infty$). In view of the condition $EX^2=1$ for $X\in L$, the sequence of the distributions of the $X_n$'s is tight. So, passing to a subsequence, without loss of generality (wlog) assume that $X_n\to Y$ in distribution for some r.v. $Y$. Moreover, the condition $EX^2=1$ for $X\in L$ implies the uniform integrability of the $X_n$'s. Hence,
\begin{equation*}
EY=\lim_n EX_n=0 \tag{20}\label{20}
\end{equation*}
and
\begin{equation*}
E|Y|=\lim_n E|X_n|\le1 \tag{25}\label{25}
\end{equation*}
Furthermore, by the Portmanteau theorem,
\begin{equation*}
p_0:=P(Y\in(-1/2,0))\le c. \tag{30}\label{30}
\end{equation*}
We will need the following lemma, which will be proved at the end of this answer.
Lemma 1: For all $X\in L$ we have $E|X|\ge1/2$.
It follows immediately from \eqref{25} and Lemma 1 that
\begin{equation*}
E|Y|\ge1/2>0. \tag{40}\label{40}
\end{equation*}
So, by Theorem 2.7 and Proposition 3.6 and in view of \eqref{30}, $Y$ has a log-concave density. So, in view of Definition 2.4., the function
$$\mathbb Z\ni n\mapsto p_n:=P(Y\in(n/2-1/2,n/2))$$
is a discrete log-concave function. So, if $p_0=0$, then either (i) $p_n=0$ for all integers $n\ge0$ or (ii) $p_n=0$ for all integers $n\le0$. So, if $p_0=0$, then either $P(Y>0)=0$ or $P(Y<0)=0$. But this conclusion contradicts the conjunction of conditions \eqref{20} and \eqref{40}. So, $p_0>0$. So, by \eqref{30}, $c>0$, as desired.
It remains to prove Lemma 1. To do that, take any $X\in L$. Let $m$ be the median of $X$, so that $P(X<m)=P(X>m)=1/2$. Let $Z^\pm$ be r.v.'s whose respective distributions are the conditional distributions of $\pm(X-m)$ given $\pm(X-m)>0$. Then $Z^\pm$ are positive r.v.'s with log-concave densities.
So, by a well-known inequality (see e.g. inequality (0.3)),
\begin{equation*}
E(Z^\pm)^2\le2(EZ^\pm)^2. \tag{45}\label{45}
\end{equation*}
Since
$EZ^\pm=2E(X-m)_\pm$ and $E(Z^\pm)^2=2E(X-m)_\pm^2$, where $u_\pm:=\max(0,\pm u)$, we can rewrite \eqref{45} as $4(E(X-m)_\pm)^2\ge E(X-m)_\pm^2$. Therefore, and because (i) $m$ is a minimizer of $E|X-a|$ in real $a$ and (ii) $EX=0$ is a minimizer of $E(X-a)^2$ in real $a$, we get
\begin{equation*}
4(E|X|)^2\ge4(E|X-m|)^2\ge4(E(X-m)_+)^2+4(E(X-m)_-)^2 \\
\ge E(X-m)_+^2+E(X-m)_-^2=E(X-m)^2\ge EX^2=1.
\end{equation*}
This completes the proof of Lemma 1. $\quad\Box$
