Moose is a red meat, similar to beef, and since beef is Paleo, you can feel free to dig into that big moose you just hunted down.
The Health Benefits
Like beef, moose is an excellent source of protein, although it is relatively low in fat. What fat it does contain tends to be less saturated and more polyunsaturated (PUFA) than the fat found in beef. This is actually less desirable, but still not a reason to avoid moose. Just make sure your moose is grass-fed (which will ensure the PUFAs are the healthy omega-3 fats, as opposed to the inflammatory omega-6 fats), cook the meat gently to avoid oxidation of the PUFAs, and you should be fine.
Mark Sisson has a whole post (written by Denise Minger) debunking the myth that eating red meat is dangerous.