In 1970, Hajnal and Szemerédi proved a conjecture of Erdős stating that any graph with maximum degree Δ admits an equitable (Δ+1)-coloring, that is, a coloring where color class sizes differ by at most 1. In 2007 Kierstead and Kostochka reproved their result and provided a polynomial-time algorithm which produces such a coloring. In this paper we study the problem of approximately sampling uniformly random equitable colorings. A series of works gives polynomial-time sampling algorithms for colorings without the color class constraint, the latest improvement being by Carlson and Vigoda for q ≥ 1.809 Δ. In this paper we give a polynomial-time sampling algorithm for equitable colorings when q > 2Δ. Moreover, our results extend to colorings with small deviations from equitable (and as a corollary, establishing their existence). The proof uses the framework of the geometry of polynomials for multivariate polynomials, and as a consequence establishes a multivariate local Central Limit Theorem for color class sizes of uniform random colorings.