No, American Sniper is not currently available on Netflix. It'll probably be a few years until the movie is available on websites like Netflix.
The film was only released on January 16th 2015, and if you look through the Netflix libraries, you'll notice that the streaming service never has movies that have been released in the past few months (unless they're Netflix original shows).
This is because Netflix has to purchase the rights to show the movies and shows it makes available to its users.
When movies like American Sniper are released, there is a limited window of time for them to charge premium rates at box offices and movie theaters.
In order to do this, the movie must remain fairly exclusive. Releasing it on Netflix would be contrary to that strategy.
Also, Netflix only costs $7.99 a month.
In order to maintain that price, they have to be very careful about which movies and shows they purchase rights to.
Here's an interesting video that explains how Netflix picks what it shows. This should help you understand why American Sniper isn't on Netflix yet...
