Are there Exchange integrals in Semiempirical methods?
Semiempirical quantum chemistry models do carry out proper Hartree-Fock calculations, including both Hartree (electron-electron electrostatics) and Fock exchange contributions to the total electronic energy. As noted in the question, the tensor of 4-center Coulomb integrals is heavily sparsified, approximated, and parameterized to simplify and accelerate semiempirical calculations. Even in the most approximate models (CNDO), the exchange interaction between molecular orbitals does not vanish in general. For example, the exchange integral between a pair of molecular orbitals $i$ and $j$ with real atomic-orbital coefficients $\phi_{i \mu}$ is simplified by the zero-differential-overlap (ZDO) approximation stated in the question as

$(ij|ji) = \sum_{\mu,\nu,\lambda,\sigma} \phi_{i \mu} \phi_{j \nu} \phi_{j \lambda} \phi_{i \sigma} (\mu \nu | \lambda \sigma) = \sum_{\mu,\lambda} \phi_{i \mu} \phi_{j \mu} \phi_{j \lambda} \phi_{i \lambda} (\mu \mu | \lambda \lambda)$.

This answer might be confusing precisely what the original question means by "exchange integral". For example, no explicit integrals are involved in calculating the Coulomb matrix elements used by most semiempirical models, and there may be some class of Coulomb matrix elements in the atomic-orbital basis that are considered to be "exchange-like" that are indeed neglected in most semiempirical models (I'm not familiar with all variants of this nomenclature). The neglect of diatomic differential overlap (NDDO) approximation that eventually superseded ZDO does allow for nonzero $(\mu \nu | \lambda \sigma)$ values when $\mu$ and $\nu$ (similarly, $\lambda$ and $\sigma$) are different atomic orbitals on the same atom.