If $f\in C[0,1]$ and $A\subset[0,1]$ is finite, can $f$ be approximated uniformly by polynomials that coincide with $f$ on $A$?

Yes.

Say $||.||$ is the supremum norm on $[0,1]$. Fix $A=\{a_1,\dots,a_n\}$. Let $I_j$ be a polynomial with $I_j(a_j)=1$ and $I_j(a_k)=0$ for $k\ne j$. Now $||I_j||<\infty$, and it follows that there exists $c=c_A$ such that given $(b_1,\dots,b_n)$ there exists a polynomial $I_b$ with $I_b(a_j)=b_j$ and $$||I_b||\le c\max|b_j|.$$

So if $f$ is continuous and $p$ is a polynomial with $||f-p||<\epsilon$ then there exists a polynomial $q$ with $q(a_j)=f(a_j)$ and $$||f-q||<\epsilon+c\epsilon.$$