Decide what type of non-academic internship you're looking for. Your non-academic internship may be related to your chosen career field--but that's not necessary. Once you decide what type of non-academic internship you want, you can start narrowing your search by contacting some companies in that field and finding out information about their internship programs.
Apply for some non-academic internships at the companies you've selected. Remember that you do not have to get approval by your school in order to get these internships, so the process can be fairly simple for the internship coordinators at these companies. Make sure you thoroughly fill out the needed application materials.
Set up appointments to visit the companies you are applying at. Visit the companies and take everything in. Listen to what the hiring coordinators have to say, take notes and look around and be interested in what's going on in the companies.
Sign the necessary paperwork with the company that offers you a non-academic internship and that you are interested in. Work with the company to set up your schedule and where you should report and to whom you will be reporting to. Take careful notes and be sure to follow through on your agreement.
Network with businesses in your industry. Attend conferences and social functions where those in your industry will be. Meet and greet those in your industry to get to know what it's like and to potentially gain an internship.
Establish the terms and payment for your internship. Many non-academic internships are non paying and allow you to set your own part-time schedule. But others offer payment and require full-time work. If your internship is non-paying, the work you do can't be as substantial legally as the work you do in a paying internship.