Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $L^p$ vector fields defined on a domain $\mathrm{\Omega }$ that is either a bounded domain in ${\mathbb{R}}^d$ or ${\mathbb{R}}^d$ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $L^p$ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals.