I have a 1990 Big Bear 350 but this questions could be for any quad I think. It's run fine for a long time but recently has been missing, mostly when under load. I found out it was running rich and changed the spark plug but still it ran poorly.
I thought it was a spark issue but what i found out was that it runs fine when the fuel is shut off from the petcock. Of course only for a couple mins then it starts to sputter. If i turn it back on it runs good for just a few seconds then bad again but as soon as i shut the fuel off it runs good. i suspected a float might have been cracked or something causing the fuel bowl to overfill but it all looks fine. The jets are clean and the float needle/seat looks fine. There is no gas in the overfill tube or blowback in the air box and it doesn't seem to be the tank vent. Any ideas? It seems really weird to me.

 

How is the float level? Blow thru the fuel line and move the float up and down. Shouldn't be able to blow thru it when the float is up. When it's running apply the chocke slightly to see if it improves, if it dose it is rich.

 

I agree with Dazey, sounds like your float is out of whack, or maybe your needle and seat is leaking by, overflowing the carb bowl, so it runs rich and sputters, shutting off the fuel lets it drain down, it runs good until the gas runs out. Start by pulling the carb and cleaning it out, make sure the needle and seat is clean, and the float is set correctly, then try to blow into the fuel inlet with the carb upside down, the float should be able to hold back any air pressure you blow, its actually supposed to hold to about 40 psi, way more than you can blow, My wolverine leaked gas, but I could not blow past the needle and seat, replaced them and it was fine after that.

 

I tried blowing into the gas inlet with fuel in the line and it holds fine but still runs the same. The needle and seat look good and everything is clean. is it possible a defective choke will let gas in when there is more pressure on the input? It would only be slight but i don't know if the choke is at all connected directly to the bowl somehow? has anyone found a choke rebuild kit? I installed a carb kit a couple years ago but the choke seemed sketchy and it didn't include any parts for it.

 

Some chokes let fuel in to richen the mix and others restrict air to richen. If your is the type that lets fuel in, you may be on the right track.