Is taking the free vaccine designated for hospital staff—which was obtained from another health unit affiliated with the Ministry of Health without official proof—considered unlawful (haram)? And is it permissible to atone for this by paying the poor instead of the health unit?

As long as the vaccine was specifically for employees of a particular unit, organizational procedures should have been adhered to, unless there was explicit or customary permission. Since you took the vaccine without permission, you can refer back to the officials to seek their permission. If they oblige you to pay its price, it should be paid to the concerned authority, and not to the poor. You must observe the organizational procedures and fulfill the contract, for Allah Almighty says: "O you who have believed, fulfill [all] contracts," and the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions regarding what is permissible."