Each candidate has probability $1/m$ of being voted for by any voter, a constant. However, each candidate's total vote count is not independent of the other candidates' vote counts, because each voter can only vote once. If a voter votes for candidate $A$, the vote cannot also be cast for candidate $B.$ This is a multinomial distribution, and the joint probability that each candidate $i$ will have $y_i$ votes is given by
\begin{align*}
p(y_1,y_2,\dots,y_m)
&=\frac{n!}{y_1!\,y_2!\cdots y_m!}\,(1/m)^{y_1}(1/m)^{y_2}\cdots(1/m)^{y_m}\\
&=\frac{n!}{y_1!\,y_2!\cdots y_m!}\,\frac{1}{m^n},
\end{align*}
since $\sum y_i=n.$ But if we assume that $n\gg m,$ we can examine each candidate's distribution as approximately binomial (and independent of the other candidates' distributions). This would enable us to use order statistics to answer the question. Under this approximation, a single candidate's distribution would be
$$f(y_i)=\binom{n}{y_i}(1/m)^{y_i}(1-1/m)^{n-y_i}.$$
Order statistics with discrete distributions are somewhat annoying - the analysis is quite different from continuous distributions. The wiki page has a good section on how to discover the distribution of the maximum order statistic. Following that procedure, we define:
\begin{align*}
p_1&=P(Y<y)\\
&=F(y)-f(y)\\
p_2&=P(Y=y)\\
&=f(y)\\
p_3&=P(Y>y)\\
&=1-F(y).
\end{align*}
Note that
$$F(y):=\sum_{x=0}^{y}\binom{n}{x}(1/m)^x(1-1/m)^{n-x}.$$
This has rather messy closed-form expression involving the regularized hypergeometric function:
$$F(y)=\left(\frac{m-1}{m}\right)^{\!\!n}\left(\!\left(\frac{m}{m-1}\right)^{\!\!n}-(m-1)^{-y-1}(y+1)!\binom{n}{y+1}\,_2\tilde{F}_1\left(1,-n+y+1;y+2;\frac{1}{1-m}\right)\!\right).$$
In any case, we compute the next three quantities regarding
$Y_{(m)}:=\max(Y_1,Y_2,\dots,Y_m)$ as
\begin{align*}
P(Y_{(m)}\le y)&=\sum_{j=0}^{n-m}\binom{n}{j}p_3^j(p_1+p_2)^{n-j}\\
P(Y_{(m)}<y)&=\sum_{j=0}^{n-m}\binom{n}{j}(p_2+p_3)^jp_1^{n-j}\\
P(Y_{(m)}=y)&=P(Y_{(n)}\le y)-P(Y_{(n)}<y)\\
&=\sum_{j=0}^{n-m}\binom{n}{j}p_3^j(p_1+p_2)^{n-j}-\sum_{j=0}^{n-m}\binom{n}{j}(p_2+p_3)^jp_1^{n-j}.
\end{align*}
To finish up, we note that the $(m-1)$th order statistic is going to be very similar - we can approximate it as the same probability as what we just calculated. We wind up with:
$$P=\left[\sum_{j=0}^{n-m}\left(\binom{n}{j}\left(p_3^j(p_1+p_2)^{n-j}-(p_2+p_3)^jp_1^{n-j}\right)\right)\right]^2.$$
