You can review the instructions from the Passport Office to see if you can meet at least one of the methods for one parent applying for a passport for a minor. The instructions are here: http://travel.state.gov/passport/get/minors/minors_834.html .
What special steps you need to take depends what documents you have now.
Because of your special circumstances, you likely should start applying for the passport several months before the trip. (I just applied for my children's passport last month and received the passports within 2 weeks. I paid for regular processing. So, the long lines people were experiencing a few months ago seems to have gone away.)