Is there a parametric distribution family for natural-numbered random vectors $X=(X_1, X_2,...,X_N)$ with a simplex-like constraint

Question

Given $a\in\mathbb{N}^N, c\in\mathbb{N}$, is there a way to generate random vectors $X=(X_1, X_2,...,X_N), X_n\in\mathbb{N}$ with constraint that $a\cdot X=c$ (inner product)? If so, how will be these distributions parametrized?