Anyone can participate in any competition as long as they are registered with the ITF and they have signed up and received a PRO IPIN. They can represent the country that the currently a legal citizen in. In the rare occurrence that the player has dual citizenship, that player can pick between either country. This always stays up to date because each player has to update their IPIN information every time they register for a tournament.
Now to explain the country permissions. Each country has a Member National Association which you first go through to gain access into playing in the ITF. Here are the regulations for the Member National Association for the USA:
- Eligibility for ITF International Team Competitions
In compliance with ITF regulations, in order to be endorsed by the
USTA into ITF international team competitions, a player shall be a
U.S. citizen as evidenced by possession of a valid U.S. passport,
Certificate of Naturalization, or other documentation acceptable to
the ITF. U.S. citizens who reside in Puerto Rico, U.S. Virgin Islands,
American Samoa, Guam, or the Northern Mariana Islands may play either
for their respective team or the U.S. international team.Once a player
has accepted a nomination for one team, the player may not compete on
the other team, unless reinstated.
USA Regulation Source