I apologize if this is in the wrong place, move it if it is. Where can I find a swing animation for my axe I modeled with Autodesk Maya? I'm making a game with Unity3D and the game requires an Axe

Unfortunately it isn't that simple.You can look around the web and find free animations, but finding exactly what you are imagining will be hard; more often you will find a attack animation and have to make do with it. Price Free/ Quality Low with a random good animation to be found.

I recommend Mixamo as it is used by a lot of indie developers. Price Free for now so get to it before it is paid / Quality High-medium to Very High.

Universities often share motion capture data made by students and teachers, it's not perfect but better quality than a amateur will make on there first few tries; also free so much cheaper than doing it yourself. If you make some interesting things with the data you can share your discovery with the University. Price Free/ Quality Low but high quality with proper editing.

Bought animations from stores will often be better quality .The Unity asset store is one such place; Unity ready also. Price Low to high /Quality Low-medium to high.

You can try doing motion capture yourself, but this approach takes months of learning. The equipment can be expensive if you want good quality but there are cheap ways of getting effective results. Price High/ Quality Depends on experience.

Learn to make animations. This is the slowest but best approach. It takes around a year to be a hobby animator and 3-4 years to be a professional. Maya is fantastic for making animations. Price medium low if you earn money as a artist/ Depends on experience.

The fastest way to get high quality animations that match your game perfectly is to hire a professional artist. Price medium although can inexperienced artist often work for cheaper/ Quality High to very high but you do get scammers.