Candidates need to have at least "a year of nappy changing experience" and an enhanced DBS check.
More than 100 "changers" have signed up in the last two weeks alone.
The website hopes to build its network to the point where parents can have someone "on-site within 10 minutes".
Richard Conway, co-founder of Childcare.co.uk, said: "Ask any parent which job they hate doing the most, and we guarantee nappy changing is top of the list.
"We knew we wanted to do something to mark April Fool's Day this year and do something that is quite obviously a joke but also fun, but we never anticipated it would take off quite like this, who would have thought that nappy changing is a genuine service that's in demand?
"Many parents have told us that they wish they could hire someone to change their children's nappies, and now they can with this service.
"We've already had more than 100 providers sign-up, but we're quite a way away from having a Nappy First Responder within 10 minutes of every household in the UK, and that's why we need even more people to get involved."