It would need to: * Survive without oxygen * Propel itself without using wings * Store enough energy for the long trip and upkeep of the below * Insulation that can handle the radiation and temperature fluctuations * Have a form of skin that can withstand the pressure change from earth to space and then the moon * Have eyes that don't pop while in space

It would need to become something literally extraterrestrial that has not evolved in the 3.7b+ years prior.

I wouldn't say it's impossible, but if evolution ever got there that creature would be so far removed from a bird that I don't think we'd recognize it :p