I started professionally when I was about your age. But back then internships were a very different thing. And so were startups and large companies to be honest. Unfortunately most of the large companies I have worked for have rules about hiring or even having people younger than 18 on premise for work purposes due to insurance and liability reasons. Many of the start ups would not hire anyone under 21. They state it is for the insurance and liability reasons but in actuality it is due to the fact that many have a "beer fridge" or go out drinking quite a bit and do not want the liability of an under aged employees.
With that said, there is hope. First off, many companies are lax in their HR departments and you might just slip through if you have the right connections. This means hustling as has been suggested: getting emails for employees, showing them your ambition and character and drive, and getting a foot in the door. This may turn into a frustrating endeavor as you might be rejected a few times, but it could work with perseverance.
Second, what makes you think you need an internship at all? You have skills, start something yourself. Maybe you want to freelance for a while, this could make for a lucrative summer experience and you will learn amazing skills that will stick with you the rest of your life. Maybe you have another app you'd like to build and grow. Again, this would be a fantastic learning experience.