Is it permissible for a non-working wife to apply for a tax refund for her third child, and does the refunded money become her property or that of her working husband, from whose salary these savings were deducted?

It is permissible for you to reclaim what you paid in taxes or insurance premiums. If the system allows your wife to request this for your third child, there is no harm in it. The money received will be yours because it was deducted from your salary. She should cooperate with you unless there is a compelling reason not to.