Are there any agencies in Fairfield County, CT that have “substitute” caregivers?

If you hire aids through a legitimate agency, then yes they will provide subs as best they can (and from my own personal experience it is not always possible because the often don't have enough aids and there's a lot of behind-the-scenes rescheduling that has to happen very fast). This is one of the benefits of working through an agency and not privately hiring aids. No one will be able to drop everything to come work at your house without prior warning unless it's a close relative and you have a special arrangement with that person.