Is the Republic of Ireland a country? Or is it a state which includes part of a country?
There is no such country as the "Republic of Ireland" and there is no soccer team fielded by the "Republic of Ireland".
"Ireland" is a State and fields a soccer team in FIFA competitions (officially called "Ireland" - "ROI" is a British media invention). "Ireland" is officially recognised as a sovereign State, has international relations and embassies, issues passports, is a member of the UN & EU etc.
I've actually done some work for a national sporting body, so I'm very familiar with the legal status of internationally fielded teams. While it is not widely advertised, for obvious political reasons, all teams that represent "Ireland" in International competition officially represent the 26 counties even if the sport is an all-Ireland organisation. [I remember being at one meeting where a high ranking official mentioned that "multinational" international teams are not allowed when asked about the prospect of an all-Ireland team].
People in Northern Ireland are generally eligible to play for Ireland as they are eligible for an Irish passport. The vast majority of International sporting organisations, including the IOC, use passport eligibility to determine nationality. The individual athlete doesnt have to travel on or even possess an Irish passport, but must prove eligibility for one.
A lot of Northern Irish athletes from a unionist background who represent Ireland actually take out Irish passports, but don't use them on a day to day basis. Some never even keep them in their possession - pick them up in person (so as to avoid letter with harp going thru NI postal system) and hand them to their sports governing body who retain them for safe keeping. Some International sports organisations, by agreement, will also allow NI athletes just to prove eligibility without having to take out the passport - documentation required varies from sport to sport and some still insist on passport only.