I rode ATV's for years, then switched to side by sides in 2010.
A mobile radio on an ATV will really depend on the radio. An amateur grade radio probably isn't going to last very long. The vibration and dust would do it in pretty quick. As N5IMS said, there are specifically designed radios for this. Actually, it's not the radio that's designed for it as it's more the control head that's designed for it.
You really need to keep the radio protected from water, mud, dust and vibration. If you look at a police motorcycle, you will usually see a large white box behind the rider. The radio "guts" are located inside there. Inside the box is a metal plate that the radio gets installed on. The plate is mounted on vibration dampeners. The box itself protects the radio. The control heads that they use are specifically designed for motorcycle mounting. Motorola, Kenwood and Vertex all make "motorcycle" control heads for their radios. They are designed to withstand the wet, dust and vibration as well as have large knobs, buttons and displays.
Drawback here is going to be the price. The other issue is that mounting the antenna can be difficult. It's not a really good idea to have a 40-50 watt radio feeding an antenna that is close to the rider. You'll often find that the public safety motorcycle radios run 15 watts or so at the most. More than that and the RF exposure is beyond OSHA rules. As for the ground plane, the radio boxes on the motorcycles usually have a metal sheet under the lid. The antennas are mounted to the box, sometimes, and the metal sheet under the lid acts as the ground plane. This works well for UHF, 700 and 800MHz, not so well for VHF or low band. You can try and use the racks, if your machine has them, but you can also use a half wave antenna, which don't require a ground plane to work. They are not ideal performers, but if you are not expecting too much, it should be fine. Getting whacked by the whip probably won't be much fun, so as they say, location, location, location.
When I was riding ATV's, I looked at all this and figured it wasn't worth the hassle. I ended up using a hand held with a longer, upgraded antenna, and keeping that in a backpack I wore while riding. Putting a handheld in a backpack with a remote speaker mic and a good antenna actually worked pretty well. Getting the radio and antenna away from body mass helps with performance quite a bit. This actually was a huge benefit for me once when I wiped out. Me and the ATV got separated, and I'd been injured bad enough that I couldn't stand. Having the radio attached to me, rather than the ATV, allowed me to call for help, rather than waiting for them to notice I wasn't with them.
As for power, yeah, right off the battery, but ATV's can be kind of noisy in the RF department. Don't be surprised if you have to add a filter to the power, or play with different grounding techniques. The small batteries and small alternators work OK as often the engines are revved pretty high. Long conversations at high power will cause issues, but hey, at least most ATV's still have a pull start (at least they used to).
For lighting, LED's are really the way to go. Reduced current draw will help the situation quite a bit, but keeping mindful about the limitations is a good idea. If there is room, a second battery can help, but it's only going to extend runtime, not prevent the issue. You are kind of limited by the size/output of the alternator.
In 2010 I happily switched to a Polaris Ranger. Having a roll cage and seat belts at least gave me the feeling of being slightly safer. A metal roof give a good ground plane for the antenna, and a bigger engine/alternator and battery help too. I have no issues running a 45 watt VHF radio with a "normal" antenna. I added a set of LED lights, and so far I haven't had any issues. Just keep an eye on your battery.
