If $n$ is sufficiently large compared to $m$, then we have
$$\frac{|\{p\mid p\textrm{ prime with }p\leq n \textrm{ and }p\equiv a\mod m\}|}{|\{p\mid p\textrm{ prime with }p\leq n \}|}\geq \frac{1}{2\varphi(m)}.$$
This is, of course, a consequence of (the quantitative form of) Dirichlet's theorem on arithmetic progressions, whose proof can be found in many textbooks. You can replace the constant $2$ by any number bigger than $1$ (at the cost of increasing the necessary lower bound for $n$ in terms of $m$).
If you ask how large $n$ needs to be in terms of $m$, that is a much harder question. Even the easier question how large $n$ needs to be to have a positive number on the left hand side is very hard. Linnik proved in 1944 that, for $m$ sufficently large and for a suitable constant $L>1$, it suffices to take $n>m^L$ for positivity. The current record on the value of $L$ is due to Xylouris who established $L=5$. (Under GRH we could take any $L>2$.)
A quantitative version of Linnik's theorem, applicable to your question, is Corollary 18.8 in Iwaniec-Kowalski: Analytic number theory. The corollary implies that, for $m$ sufficently large and for suitable constants $L>1$ and $c>0$, we have, for $n>m^L$,
$$\frac{|\{p\mid p\textrm{ prime with }p\leq n \textrm{ and }p\equiv a\mod m\}|}{|\{p\mid p\textrm{ prime with }p\leq n \}|}\geq \frac{c}{\varphi(m)\sqrt{m}}.$$
The proof allows to specify $L$ and $c$ explicitly, but I don't think this was carried out anywhere. At any rate, Iwaniec-Kowalski express $L$ in terms of four constants, contained in Principles 1-3 on page 428, for which explicit values can be found in the literature. That is, for the above quantitative version of Linnik's theorem, an explicit value of $L$ could be given with little work.
P.S. Your question has nothing to do with Chebotarev's density theorem, except that Chebotarev's theorem implies Dirichlet's theorem (in various forms) as a special case.
