From what I remember, Wonder Woman couldn't fly, but now that I think about it, I watch Justice League the animated series and she does have the amazing ability of flight.
A little research showed that Hermes granted her flight, but I couldn't find the circumstances or why she was given flight later (or when for that matter).
In the Golden Age (from the early forties to the late fifties/early sixties), the character couldn't fly at all.
In the Silver Age (late fifties/early sixties to the early eighties), Diana gained the ability to glide on air currents starting in Wonder Woman (vol 1) issue #98, May 1958.
Post-Crisis on Infinite Earths (in 1985), Wonder Woman's origin has been rebooted, and she could fly freely from the start. Hippolyta made a baby out of clay and the goddesses (plus Hermes, for some reason) granted Diana beauty, power, strength, speed and flight.
In 2011, DC Comics' entire line of publications was relaunched, and Wonder Woman's origins have been rebooted again. This time she was a demigoddess, daughter of Hippolyta and Zeus, and was also gifted the ability to fly by Hermes. In this case Hermes scratched her with one of his feathers as Wonder Woman was trying to save Zola who was falling to her death.
Wonder Woman first gained a Flight-like power as early as Issue # 98 of her original comicbook series. They referred to it as the "Power to Glide on the Air Currents". This ability became a regular part of her power set throughout the 60s, 70s and 80s. When the Lynda Carter Wonder Woman TV Series began in 1975, the producers decided to base their rendition on the original golden age WW comics where WW did NOT have the ability to fly on her own.
When the WW Comicbook was rebooted and renumbered and had a new issue # 1 in 1986, The Greek God Hermes was shown giving Diana the Power of flight when she first came to life. In 2012, the 4th volume of her comicbook series, during a scene where Diana's friend Zola was thrown off the cliff of Mt. Olympus, the God Hermes threw one of his magic feathers like a dart at Diana's leg and immediately after gave Diana the Power of Flight which allowed her to catch Zola.
In her current series (Volume 5), Wonder Woman was given her special powers by the Amazons' patron Gods AFTER she arrived in Man's World. Among those powers was the power of flight which Diana discovered she had during a battle with "The Sear Group" (Disciples of Ares, God of War)
2 possibilities:
She was born on Themyscara and basically a demigod and she was trained by women soldiers making her able to fly because of her birthplace and heritage
Themyscara's gravity is stronger than Earth's and when she's on earth she can fly because of Earth's gravitational orbit making her stronger and better and maybe, possibly able to fly, but I could be wrong.
