Can I petition the court for a private mediator for custody and visitation issues

You may have to wait until a mediator is actually appointed to then file a complaint against him/her on a concern of conflict of interest. See below from the California court system on child custody mediation. http://www.courtinfo.ca.gov/selfhelp/family/custody/custmediation.htm#complain

Private mediation is generally a matter of choice, including who is selected. If you are referencing Court ordered mediation through the conciliation office of the Court system, then, regardless of what you CAN do, you must consider what you should do. You must consider many factors, including the number of mediators, the nexus between such a relationship with some mediators and other mediators, choosing the real battles, if any, the trickle down effects to the children for which these services are obtained, the fact that prior familiarity itself is not per se a disqualifying factor, the option for private mediation, and a host of other factors. Presumably, subject to your particular facts, I would say DON'T DO IT.

Not successfully. While you may be able to disqualify a mediator who actually knows your ex-wife, (and a mediator who knows her well will probably disqualify herself) you can't disqualify all the mediators in the county who may simply know someone who knows her.