You can find all the information about urgent medical aid, the Family Doctor's Advice Line, the crisis support telephone number 116 006 etc on the website https://kriis.ee/en/security-situation-europe/ukrainian-war-refugees/staying-estonia.
Medicines can be bought from a pharmacy with a paper prescription issued in Ukraine, except for narcotic and psychotropic drugs. It must be clear to the pharmacist which medicine is prescribed for the person.
War refugees arriving from Ukraine can turn to family medicine centers with health concerns (prescription extension, chronic diseases, other health concerns), which is listed on the Health Insurance Fund's website: family doctor centers.