Is there an API to get real-time FAA flight data?
You can't get real-time data from the FAA without an operational need, and organizations which do receive real-time data cannot legally re-distribute this data publicly, except to other organizations approved by the FAA. An operational need pretty much means you have to be a flight dispatcher for an airline or commercial operator, not just have an interest in tracking flights.

You can get some real-time data from your local area using an ADS-B receiver. See this  question for more information: Where can I find real-time flight tracking with the least amount of delay?.

If you are okay with the five minute delay, then flightaware has an API you might consider. The direct feed from the FAA is called Aircraft Situation Display to Industry (ASDI) and I believe you can apply for the delayed feed, though I've never gone through the process, and the government's site is very sparse in information. I would suggest contacting them before attempting to apply if it's something you're interested in.