In this article we give a combinatorial formula for a certain class of Koornwinder polynomials, also known as Macdonald polynomials of type $\tilde{C}$. In particular, we give a combinatorial formula for the Koornwinder polynomials $K_λ = K_λ(z_1,\dots,z_N; a,b,c,d; q,t)$, where $λ= (1,\dots,1,0,\dots,0)$. We also give combinatorial formulas for all ``open boundary ASEP polynomials'' $F_μ$, where $μ$ is a composition in $\{-1,0,1\}^N$; these polynomials are related to the nonsymmetric Koornwinder polynomials $E_μ$ up to a triangular change of basis. Our formulas are in terms of rhombic staircase tableaux, certain tableaux that we introduced in previous work to give a formula for the stationary distribution of the two-species asymmetric simple exclusion process (ASEP) on a line with open boundaries.