There are two I could find:
Tokyo Anime Award: The Tokyo Anime Award started from 2002, but was named in 2005. The
first, second and third awards were only named simply
'Competition'.1 The award ceremony was held at the Tokyo
International Anime Fair (TAF) until 2013. In 2014, after the merger
of Tokyo International Anime Fair with Anime Contents Expo and
formation of AnimeJapan convention, Tokyo Anime Award started as a
separate festival called Tokyo Anime Award Festival (TAAF).
As per @senshin's input: They do consider western animation as anime too for the Awards. Flutter was the winner of Open Entries/Competition Grand Prize in 2007 being the first non-asian entry*.
They also have a specific website on it: http://animefestival.jp/en
There is one more from America:
American Anime Awards: The American Anime Awards were a series of awards designed to
recognize excellence in the release of anime and manga in North
The first and, as of 2015, only annual American Anime Awards balloting
was supervised by Milton Griepp of industry website ICv2. The first
gala awards presentation was hosted in New York City on February 24,
2007 at New York Comic Con. The hosts of the evening were eight
actresses from the anime production company ADV Films: Christine
Auten, Shelley Calene-Black, Jessica Boone, Luci Christian, Alice
Fulks, Hilary Haag, Taylor Hannah and Serena Varghese. A streaming
version of the one hour awards ceremony can be seen on IGN.com. The
awards were later broadcast on the Anime Network.