Can I give our long-term care residents zoster vaccine, injectable influenza, and pneumococcal vaccines on the same day?

Yes. CDC’s “General Best Practice Guidelines for Immunization” advise that non-live vaccines, such as Shingrix, can be administered concomitantly, at different anatomic sites, with any other live or non-live vaccine, including the vaccines you listed, as well as COVID-19 vaccines. They should be given as separate injections, not combined in the same syringe.