Should I ask for a letter of reference to take with me?
No. Not in most fields, in most locales.
Letters of reference have almost no value with the hiring managers I know, including me. It's just a piece of paper. Far better is to secure someone as a reference who can be called when needed, and who will say good things about you.
If you have people your work with or work for who like you and would be good references, casually ask them if they would be okay being a reference. Get their contact information for future use.
Then, when you have gotten to the point in a job search where you need a reference, contact them again. Give them the specifics of the situation - the job, the company, the name of the person who would be checking the references, etc. And thank them for their help.