My guess is that they are but I can't find any information in my books.
For $\mathbb{R}^{\mathbb{R}}$, Willard describes the basic open sets as all functions whose values are within $\epsilon_k$ on $(1,2,...,k,...,n)$-many fixed coordinates, and so it would seem that I can find a polynomial in any such given open set.
For the latter ($\mathbb{R}^{\mathbb{N}}$) I am more unsure.
Here's a proof that $\mathbb {R}^{\mathbb {N}}$ is separable. For $F$ a finite subset of $\mathbb {N},$ let $D_F$ be the set of real sequences $x_n$ such that $x_n\in \mathbb Q$ for $n\in F,$ and $x_n = 0$ otherwise. Then $D_F$ has the cardinality of $\mathbb Q^{F},$ so is countable. Now the set of all finite subsets of $\mathbb N$ is countable, so $D = \cup D_F$ is countable. If $U$ is a basic open set in $\mathbb {R}^{\mathbb {N}},$ then $U$ is the countable product of open sets in $\mathbb R,$ where each of these open sets is $\mathbb R$ itself except for a finite subset $F$ of $\mathbb N.$ Thus $U$ will contain an element of $D_F,$ hence of $D,$ and that does it.