What is a person (not a company) who provides a service called?

If you employ a person and pay them a regular wage or salary, they are an employee.

If you negotiate a contract with an individual to provide a service, with that person billing you for services rendered, that person is a contractor.

If you hire a company to provide you services, the people working for that company are employees (or sometimes contractors) of that company and contractors from your viewpoint.

The general term for any of the above is "worker".

The correct word for someone who provides a service is servitor, although this is a bit archaic in usage and usually they would be referred to according to exactly what service they were providing.

In general, a person who provides a professional service is called a consultant.

It's still called a company, even if it's only one person.

If you want a more specific word, try "sole proprietorship".

The word that you are looking for is supplier which is defined as:

someone whose business is to supply a particular service or commodity

So in this case it would be service. In fact you could also use simply provider, which in fact is a synonym of supplier.