Он не поддерживает ни Object.values, ни Array.includes.

CryNet,
пара вариантов

const getValue = _ => 'zOo';
let toArrayStaff = [{'full_names' : { foo: "bar", baz: 42 }}, {'full_names' : { foo: "Zoo", baz: 42 }}];
let search = getValue("search").toUpperCase();
//toArrayStaff = toArrayStaff.filter(element => Object.values(element.full_names).map(value => value.toString().toUpperCase()).includes(search));
toArrayStaff = toArrayStaff.filter(element => Object.values(element.full_names).some(value => value.toString().toUpperCase()===search));
console.log(toArrayStaff)