The bus between Dubrovnik and Medjugorje is scheduled to operate all year around, travel time for the 145 km long route is about 4 hours. Alternatively it is possible to travel from Dubrovnik to Mostar and then take bus from there to Medjugorje.
The bus does not departure from the Dubrovnik airport, only from the bus station. From the Dubrovnik airport you have option to arrange for transfer for any number of persons, you can see transfer prices here.