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".

In general, grievance (from class. lat. gravis: heavy) is a wrong or hardship suffered, real or supposed, which forms legitimate grounds of complaint. : a feeling of having been treated unfairly [noncount] He has a deep sense of grievance against his former employer. [count] She has been nursing a grievance all week. 2 [count] : a reason for complaining or being unhappy with a situation In the petition, the students listed their many grievances against the university administration. 3 [count] : a statement in which you say you are unhappy or not satisfied with something The union has filed a formal grievance [=complaint], accusing the company of unfair labor practices. Several customers came to the front desk to air their grievances.

The previous answers are both correct. However, the basic difference is that 'complaint' is an action corresponding to the verb 'complain': you have a complaint when you would like to complain to somebody about something that has affected or affects you negatively. A grievance is a state (which has very little to do with the verb 'grieve' which is mostly used for feelings regarding the loss of a loved one). In other words, complaints need to be brought up in a conversation but grievance is an emotional state towards something that affects you negatively and that you perceieve as wrong and it doesn't require communication with others.