Approximating PDEs on surfaces by the diffuse interface approach allows us to use standard numerical tools to solve these problems. This makes it an attractive numerical approach. We extend this approach to vector-valued surface PDEs and explore their convergence properties. In contrast to the well-studied case of scalar-valued surface PDEs, the optimal order of convergence can only be achieved if certain higher-order relations between mesh size and interface width are fulfilled. This difference results from the increased coupling between the surface geometry and the PDE for vector-valued quantities defined on it.