Yamaha grizzly 350 Starting up the quad, it stalled. Opened the air filter and noticed the rubber hose next to the filter was an oily substance spewing from this hose. Any idea what this could be?

There isn't one. You have to put one in-
line

gas in the air box is usually cause by dirty and/or stuck carb float

Disconnect the fuel line at the fuel filter or carby, without starting the engine, turn the key to the on position. The pump should start , now watch to see if you have an even flow from the hose, if it looks like spluttering you may have a problem, also check the fuel filter.

what year? pull cable under seat on some? (under plastic ) has looped cable , just pull it. along edge of seat on others small lever pull up towards seat.

Sounds like it has sat for a while and your jets are gummed in the carb there is a you tube video on this topic that will walk you through the repair other thing to check turn fuel tank valve to off position remove fuel hose at carb slowly turn valve on to see if you have fuel to the carb if there is fuel there turn valve off install hose back on carb remove remove hose at tank valve now on the bottom of the carb you will see a plug or screw loosen this ( there may be fuel running out of drain hose have a bucket to catch fuel) now on the hose at tank valve blow threw it ( do not use air compressor) air should blow freely if it dose not your needle valve is stuck and there is no fuel getting to the carb or stuck float or a hole in the float and full of fuel

First check the fuses.check the start ,run switch,there should b some kind of protection for this..possibley a in line fuse??

These quads have a thermostat-controlled radiator fan, which won't turn on until the engine is hot. Perfectly normal. Also, if the fan comes on, make sure you use the kill switch to turn the quad off, rather than just turning the key. This allows the engine to cool properly and will save your engine gaskets and other parts. It could take quite a while to cool down...my 2009 Grizzly sure does.

Check the starter motor on the bench clamped in a vice ,you may have sluggish or worn brushes .

Check the carb boot were it attaches to the cylinder head and see if can notice any cracks or hole in it.

is this bike a yamaha rd 350 twin cylinder 2 stroke? if so, its air cooled and you should not be getting a ltr of water.. but it is common for them to have a "oily" fluid come out the exhaust because it is a 2 stroke. but not alot. it comes from the engine not burning all the oil gas mix. does the engine spit and sputter? make sure there is no water in the tank, make sure the air filter is clean and buy hotter spark plugs. also adjusting the fuel air mixture may help. hope this helps...