EORI numbers are mostly used when shipping commercial items. However, some countries will require you to have an EORI number when shipping personal effects so it is always best to check with the destination's local customs authority prior to booking to query whether you will have to obtain one.
You can find details on how to apply for an EORI number by searching for the application form online, please see below a list of links to various countries and with guidance on how to apply:
Bulgaria - https://www.lawyers-bulgaria.com/eori-registration-in-bulgaria
Czech Republic - https://www.czech-lawyers.com/eori-registration-in-czech-republic
Latvia - https://latvija.lv/en/PPK/finanses-un-nodokli/nodevas-nodokli/p812/ProcesaApraksts
Lithuania - https://www.onze.lt/en/services-provided/registration-of-the-eori-code/