This of course varies from project to project, but yes: there are both businesses and freelancers who do this. But more so for large free software projects, than for small ones.
As an example, look at a Drupal. Drupal is a FLOSS project (requiring all distributed components to be licensed under GPLv2+) that also sustains a large community of individual developers and companies that can help you with adapting and updating any Drupal project.
When I am approached by clients about paid work for Drupal, the client is almost always aware of this, and tell me up front that my company is just one of the service providers they consider. They expect my offer to be competitive compared to other companies that offer Drupal-related services. And if they know what they're doing (not all companies understand this), they require us to distribute what we deliver to them under GPLv2+ - which means that they can kick us out and hire somebody else to work on the software we create for them if they're are dissatisfied with our services.
I also believe that this is the case for very many other free software projects, but my first hand experience is with the Drupal ecosystem.