Let $X$ be a set of variables, and consider the polynomial ring $K[X]$ in the variables $X$ over a field $K$. Let $Mon(K[X])$ denote the set of monomials of $K[X]$. Let $M$ be some monoid acting on $Mon(K[X])$ such that $\rho1=1$ and $\rho(ab)=\rho(a)\rho(b)$ for all $a,b\in Mon(K[X])$ and $\rho\in M$. How do I formally "extend" this action to a monoid action on all of $K[X]$? I want to do it in a natural way, so that things like $\rho(f+g)=\rho(f)+\rho(g)$ and $\rho(af)=a\rho(f)$ hold for all $f,g\in K[X]$, all $a\in K$ and all $\rho\in M$. Essentially I am starting with a monoid action on monomials and I wish to "linearly extend" the action to the entire polynomial ring.
My question is, is there a well-known formal way of doing this? How do I formally define the action on $K[X]$?
I thought about doing something like defining $$ \rho\left(\sum_ic_im_i\right)=\sum_ic_i\rho(m_i) $$ where the sum is finite and $c_i\in K$, $m_i\in Mon(K[X])$ and $\rho\in M$.
Your construction is the way to go.
One sidenote: ${\rm Mon}(K[X])\cup\{1\}$ is basically the free commutative monoid on the set $X$, so the original action $\rho$ is already determined by its restriction to single elements of $X$, and that can be an arbitrary function $X\to{\rm Mon}(K[X])$ (which all can be directly seen right away).