I do not see clearly how to use the information that the coefficients be integer numbers, apart the fact that $1\le |b_m|\le |a_n|$. The following is an explicit bound, that also holds for complex coefficients.
By a classic estimate on the zeros of a polynomial, all (complex) roots of $f$ are bounded in modulus by $$1+\max_{0\le i < n}\frac{|a_i|}{|a_n|}\le M+1\\ ,$$ because $|a_m|\ge1$. Since $b_i$ is the $i$-th elementary symmetric polynomial of some $m$ of the roots of $f$, times $b_m$, which is less that $M$ is size, we have
$$|b_i|\le M {m \choose i}(M+1)^i\\ .$$