I went from programming intern to gameplay programmer at ArenaNet. My application consisted of my college projects and the tutoring that I did in college for CS. What I would suggest is applying to a lot of different places. Going through multiple interviews with different companies to learn the interviewing process can be more helpful than you'd initially think, and you might find that you'd prefer to work with a different company. What you want for the programming intern position is experience in design patterns, mathematics, debugging skills, as well as knowledge about programming fundamentals.
I recommend programming in your off-time and building small projects that do small things. Make sure you're able to explain what you write, how it works, and why you choose to approach problems the way you do. These portfolio pieces show what you are actually able to do, not just what you know about, and are essential for showing that you're good for the position.