In the game Spore, is there a non-cheat way to neutralize the Grox, whether by allying with them, utterly defeating them, or some other method?
You can both ally with them, or utterly defeat them.
Allying with the Grox is not taken lightly in the Galactic Community, and will result in upsetting all spacefaring nations.
To be able to utterly defeat them, you will need an official mod/patch (use save-as) and have gained secret at the center of the universe. Also, This will take some time, as they have have 2400 systems (a total of about 5000 planets) at the start of the game.
more details (including spoilers and exact methods) can be found at the Grok entry in the Spore Wiki
There are three ways you can kill all the Grox (though all 3 of them are very time consuming).
Buy out all the Grox planets using trade routes.
Improve the T-Score of all their planets to at least one (They will start a war with you, but they are at war with you when you first meet them, so. . .).
Destroy all their planets using military force (just like #2, they will start a war with you, but of course, they probably already are).