There are a number of resources for determining the mailing addresses of the judges to whom you will be applying. The best on-line resource is the "Clerkships" section of the Symplicity system (http://law-wisc-csm.symplicity.com/students). In that section, you can export into an Excel file the names and address of those judges to whom you wish to apply. Then you can perform a mail merge of those names and addresses into your cover letters and mailing envelopes. (See instructions on how to perform a mail merge.)
For federal judges, you can also use hard copies of either the Almanac of the Federal Judiciary (located in the law library) or the Judicial Yellow Book (located in the law library and in the Career Services Office) to find judges' contact information. Because these resources are only published twice per year, if you need more up-to-date information (e.g., for newly-confirmed judges), you should check the web site of the court in which the judge sits (most of which are accessible at www.uscourts.gov). If you still cannot find a judge's address, please contact the Career Services Office, where the judicial clerkship advisor has access to the internal database of the Administrative Office of the U.S. Courts.
For state appellate judges, you can also utilize the Judicial Yellow Book. For other state judges, you can access state court web sites at www.courts.net or www.ncsc.dni.us/courts/sites/courts.htm (National Center for State Courts).