Can I create an Xbox avatar if I don't own an Xbox 360?

You can create an XBox Live account at http://live.xbox.com, even if you do not own an XBox/XBox 360, and you can edit many options of your XBox Live profile. However, there currently is no way to create/change your avatar without an XBox 360 w/ an internet connection.

Do any of your friends have an xbox 360?

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      If yes, you can ask one of your friends to let you create an account on their xbox, and create your avatar. Then you can use that account on xbox.com, and see your avatar there. You will also be able to remove that account from your friend's system, and just view it on xbox.com for now, and move that account to any xbox in the future.

Speaking as of 2016,

1) You can change your avatar from your Windows Phone 8 and Windows 10 devices:

Using the Xbox Avatars app, you can change your avatar from your PC or phone.

2) You can change your avatar on Xbox.com:

You can change your avatar online too! However, it requires Silverlight, which you can download here.