Edit: This is an answer to the wrong question. This explains the finite generation of $M_{k}(\Gamma_{1}(n), R)$ as an algebra, not as a module over the graded ring of level $1$ modular forms.
This result is already known. One good source for this is the paper of Nadim Rustom (Generators of graded rings of modular forms, published in the Journal of Number Theory in 2014, the arXiv version is here). This paper shows more, that the graded ring of modular forms for $\Gamma_{1}(n)$ is generated in weight at most $3$ for $n \geq 5$. This is fairly simple over $\mathbb{C}$ and follows from a lemma (dating back to Mumford, essentially) about surjectivity of the map $H^{0}(X,\mathcal{L}_{1}) \otimes H^{0}(X,\mathcal{L}_{2}) \to H^{0}(X, \mathcal{L}_{1} \otimes \mathcal{L}_{2})$ where $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are two line bundles on a curve $X$. However, it takes a bit of work to translate this result into the desired statement for an arbitrary commutative $\mathbb{Z}[1/n]$ algebra.
I would expect that there should be a much simpler way to see the finite generation of $M_{k}(\Gamma_{1}(n), R)$, but I don't know one.