I brought up the wiki page for Equestrian Olympic sports and saw quite a few small countries on there that I had no idea ever competed (Latvia, Liechtenstein, Palestine, etc). It appears most were represented by a single individual at one point (vs having a whole team). It got me wondering how those individuals go about qualifying - I assume that some countries don’t have their own NGO for equestrian sports, which aren’t those who is responsible for “electing” (choosing/ assigning/ etc) those who attend and create qualification standards?
The FEI sets the eligibility requirements and qualification standards for Equestrian events at the Olympics but they have to be approved by the IOC. National Federations choose which of their eligible athletes to send. The FEI lists 136 countries that have National equestrian Federations, so it is probably more than you think.
Link to the FEI rules for Tokyo, including qualification procedures:
List of National Federations in the FEI database:
The rider does not actually have to live in the country. Qualification does not have to happen in the home country. There are rules about citizenship of the horse owner, which does have to be the same as rider, otherwise citizens can be flung far and wide. I am sure many riders may be based in Europe and use the FEI events to meet the standards.
I believe in most sports there is an allocation of participants slots to allow for the participation of “smaller” countries. I know in archery there is a quota system that has spaces for a certain number of athletes from smaller population countries.