Say we are given a sequence of integers $\{a_i\}_{i \in \mathbb{N}}$, as well as a pair of integers $n, m$. How can we show that there always exist positive integers $s, r$ such that the sums
$a_s + a_{s+1} + \cdots + a_{s+r -1}, \\ a_{s+r} + a_{s + r + 1} + \cdots + a_{s + 2r - 1}, \\ \quad \quad \quad \quad \quad \quad \vdots \\ a_{s + (m-1)r} + a_{s + (m-1)r + 1} + \cdots + a_{s + mr -1}$
are all divisible by $n$.
My first instinct is to use van der Waerden's theorem, perhaps more than once, in order to make the argument work, but I'm having trouble making this argument rigorous. Does this approach even work?
Your idea is actually correct. Just consider the sums $S_k=a_1+a_2+....+a_k$ and color all integers in $n$ colours depending on their remainder modulo $n.$ More precisely, colour of $k$ is determined by $S_k\bmod n.$ Van der Warden then tells you that there is an arithmetic progression $S_s,S_{s+r},S_{s+2r}...S_{s+(m+1)r}$ with the same remainders modulo $n.$ Substract the latter to get the desired result.