Approximation theory on the disc

The uniform limit of a sequence of complex polynomials is a holomorphic function. To get beyond holomorphic functions you need to work with polynomials depending on both $z$ and $\bar{z}$. Check this older MO question which deals with uniform approximations on the square via multivariate Bernstein polynomials.

The disk case can be reduced to the square case as follows. Given a continuous function $f(r,\theta)$ on the disk $r\leq 1$ we can extend it to a continuous function $F(r,\theta)$ in the plane by setting

$$ F(r,\theta):=\begin{cases} f(r, \theta), & r\leq 1,\\ f(1,\theta), & r\geq 1. \end{cases} $$

Next approximate $F$ on $[-1,1]\times [-1,1]$ using the bivariate Bernstein polynomials as detailed in the linked MO post.