I was hypothesizing an organism that could create it's own fire with the use of a flamethrower-like organ. I was thinking that this animal has a oxygen-free organ that hold diethylzinc and when the animal feels threatened or such, its sprays out the diethylzinc with instantly ignites when in contact with air, practically acting as a flamethrower. I mainly wondering if this seems like a plausible scenario and if diethylzinc would be a good substance to use.
Mix the diethylzinc with oil.
Oil (think whale oil, or spermaceti) can be produced using biology with no problem.
A hydrophobic light oil would also tend to exclude oxygen. The oil reservoir would be a safe place to store your diethylzinc igniter molecules.
A stream of viscous oil would be an easy thing to use as a breath weapon.
Mixing the diethylzinc with the oil would mean on contact with air, not only the diethylzinc ignites but then it also ignites the oil. You get a lot of added value from the zinc - once the zinc oxidizes it is gone, but when the oil ignites it will burn fiercely.
Your general idea is definitely possible and the use of Diethylzinc is a creative and possible one.
Nevertheless I would go an easier route as this is the one evolution often takes. Diethylzinc combust with contact to oxygen which means that you would need a bladder like organ to produce and store your fuel without having any Oxygen in this organ (also not in the cells of the organ). Therefore I would go with a fuel which does not self-ignite.
The first think which comes to mind is Methane as this is something a lot of animals are already exceptionally good at producing. Now that you need a source of ignition I would propose electroplaques to create a biological spark plug. Your average electric eel does "only" produce about 1 Ampere at 500Volt but you could give your flame thrower animal just a bit more power to get to the Voltage you need to ignite fuel.
Now you have a safe flame throwing animal which produces it's fuel as a byproduct of it's digestion. This is an efficient solution which, to me, sounds like something evolution would like.
If you want to precisely figure out how much you need to ignite Methane, here is a paper about the ignition of methane-hydrogen-air mixtures (which also examines the ignition energy of methane-air mixtures).
Nature is ahead of you -- the bombardier beetle uses a mix of hydroquinone and hydrogen peroxide to similar effect. And it works!
Personally, I like your idea of Diethylzinc. According to wikipedia:
Diethylzinc (C₂H₅)₂Zn, or DEZ, is a highly pyrophoric and reactive organozinc compound consisting of a zinc center bound to two ethyl groups.
Now, this is the case in STP conditions (I think), so if your dragon was to go around doing burning people, it would have to do it in standard conditions. I think the main thing is that the dragon must somehow maintain its internal physical conditions. Basically its fire-chamber's temperature, pressure, chemical balance has to be very regulated, especially if your dragon wants to set things on fire in deserts and tundra. Otherwise, this power would be localized to areas where conditions allow it to, and nowhere else.
My only critique of Diethylzinc is where it's found: in the real world it must be synthesized, and is often found commercially as a solution in hexanes, heptane, or toluene, or as a pure liquid. So unless you can figure something else out, your dragon's fire chamber has a lot in common with an oil tanker or fracking operation.
EDIT: Bombardier beetles use both a fuel, hydroquinone, and an oxidizer, hydrogen peroxide. A pair of organs to produce and store the fuel and oxidizer, plus a third organ to catalyze their mixture and safely direct the resultant plasma, would make sense to get around diethylzinc's reliance on standard temperatures and pressure.
Now the limitation here is that although this could help, the beetles themselves release something that feels like hot, boiling water, which could change the nature of the fire being produced by your dragon. So, maybe copy the beetle's containment system, but definitely change the chemistry inside. Or, mix the two and you can effectively construct a more efficient way to set fires as a dragon.
Source: @Ghedipunk's comment
I'd personally go with a binary that produces an exothermic reaction (like the bombadier beetle) rather than a single pyrophoric material. That gives you a safety margin; you could imagine the two reservoirs of material have individual orifices that each spray their respective chemicals out so they only ignite once they mix outside of the animal.
For instance, potassium permanganate will ignite when mixed with glycerine or ethylene glycol (in fact, it's sold in survival kits as a firestarter). If you have the potassium permanganate in an organic oil or water suspension, spray that and spray the other chemical along with it, the fine mist should create a nice fireball.