You don't apply for mediation, you agree with the other party that you want to hire someone to mediate the dispute. If an agreement is forthcoming from the mediation, the agreeement can be reduced to writing an filed with the court.
Contact a community dispute resolution center, located in most larger communities, and you might find one that offers their services on a sliding scale.
Google "Pierce County Center for Dispute resolution" and talk to them. They will give you all the information you need to apply. Hope this helps. Elizabeth Powell