I think the answer is yes, because $W$ must be isomorphic to $C(K)$ for some closed subset $K\subset [0,1]$. The condition about idempotents implies $K$ is connected. The only connected subsets of $[0,1]$ are closed sub-intervals, and these are either homeo to $[0,1]$ itself or are degenerate singletons.

EDIT: the following answer was based on the assumption that $W$ is Banach and the ring morphisms $R\to W\to {\mathbb C}$ were continuous. My apologies for misunderstanding the question.

I think the answer is yes, because $W$ must be isomorphic to $C(K)$ for some closed subset $K\subset [0,1]$. The condition about idempotents implies $K$ is connected. The only connected subsets of $[0,1]$ are closed sub-intervals, and these are either homeo to $[0,1]$ itself or are degenerate singletons.