The authors study finite element approximation methods for some types of second-order elliptic eigenvalue problems (EVPs) for vector-valued functions on a convex polygonal domain in the plane, with nonstandard boundary conditions (BCs) of nonlocal type. They argue that similar convergence results and error estimates hold as those established for elliptic EVPs for a scalar function, with classical local BCs of Dirichlet, Neumann or Robin type. Here, the nonlocal character of the BCs constitutes a major difficulty in the analysis, requiring the introduction and error estimation of a new, suitably modified (vector) Lagrange interpolant on the finite element mesh. The theoretical error estimate for the eigenvalue is confirmed by an illustrative numerical example.