To "make a complaint about" / "to complain about" (not "for") is the more general expression. You can complain in all sorts of situations, formal and informal.
"to have a grievance about" something is a little different. You have a regular problem with someone or something which is not easy to resolve. It can be informal, say in a family, or also formal, especially in an employment context. A grievance arises in the context of a regular relationship.
In your context, "complain about" is the best choice as the situation is not personal enough to justify the use of "grievance".