In this paper, we provide a simple criterion for the Schubitope $\mathcal{S}_{D}$ associated to a diagram $D$ to be lattice-free. We further show that $\mathcal{S}_{D}$ is lattice-free if and only if its Ehrhart polynomial is equal to the product of Ehrhart polynomials of the Schubert matroid polytopes corresponding to each column of $D$. As applications, we obtain that the Newton polytopes of the Schubert polynomial $\mathfrak{S}_w(x)$ and the Grothendieck polynomial $\mathfrak{G}_w(x)$ are lattice-free if and only if $w$ avoids the patterns 1423, 1432, 13254, and confirm several conjectures by Mészáros, Setiabrata, and St.Dizier on the support of Grothendieck polynomials for this class of permutations.