I was wondering this too. I doubt I could have one delivered in time to participate 

you could always get someone else to test your game, but it's easier if you have your own console to push builds to. I'm pushing builds all the time as I develop.

The game has to work on OUYA to qualify, so it's best if you have your own console so you can test it easily, as Toasterkidd said. You can probably find someone else to test it, but you might be waiting hours just to hear back that the game isn't working, which is a lot of time lost.

OUYA can be a bit of a fickle beast, so it's a good idea to own one to make sure your game runs on it.

How can you get started without one?

1. OUYA is an Android device, it runs Android Jellybean (4.1) which is API level 16, it is 32-bit. You should make sure that any game development IDE you use supports that. I believe Unity 2019.2 and older might support this, but you should double-check. Game Maker Studio 1.4 has decent OUYA support, and Game Maker Studio 2.2.5 can still export to OUYA, however Game Maker Studio 2.3 and above appears to make this impossible. At least I can't get it to work. App Game Kit Classic also appears to work I've heard. And I believe older versions of Godot too.

2. The OUYA is a low-end device by todays standards, so make sure you don't use anything too intensive in your game, such as dynamic lighting or full-screen shader effects, lots of transparency etc. These can cause performance issues, so you're best keeping them to a minimum. Perhaps consider designing your game around a 1280x720 resolution too.

3. Look into how your choice of IDE handles the OUYA Controller, some of them don't offer great support for it, but you can probably get started adding support for it without owning an OUYA, and it might save you some time when you eventually get one.