This website contains a chart of damage for military and improvised explosives and other weapons in nWoD, including a "civilian" flamethrower, although I agree with what SevenSideDie said in the comments—an implement able of shooting jelly fuel is as good as a military grade flamethrower and should deal the same damage.
Alternatively, if we define the word to mean any weapon able of shooting aimed flames, flamethrowers don't necessarily require jelly fuel—in fact, commercial ones use propane. In a modern setting, one can be somewhat easily built using propane as fuel or more crudely improvised using gasoline in a water gun, which is not as potent as a military flamethrower, but still fits the description.
To calculate the damage, you can use the rules in the same website for a molotov cocktail as a base. Molotov cocktails are arguably made with gasoline, although this is not a rule, different fuels can be used to make the flame burn hotter or "stick" to the target. The flame of burning petrol is around 500°C.
I think it's reasonable to assume that the base damage of a fire-based weapon (that is, the damage dealt while the target is still under the flame, not simply on fire) is proportional to the temperature of the flame. Therefore, once the base damage for a gasoline-fueled weapon is established, we can just use that to calculate the value for other flames. Petrol flames are relatively low compared to other fuels, so different fuels can be used for higher damage. There are a few reasonable candidates for a homemade flamethrower:
Propane cylinders are easy to find in camping stores, and the flame is around 2000°C. If combined to an oxygen cylinder, which can probably be purchased from medical stores for treatment of lung conditions, or stolen from hospitals, the mixture would burn at around 2500°C.
Butane can be purchased as lighter fluid, and also burns at around 2000°C in air. Curiously, I haven't been able to find data regarding its flame temperature when mixed with oxygen.
Acetylene is an even better candidate: it burns at 2500°C and can be mixed to oxygen for a 3500°C flame. It is used in welding, so probably easy to obtain as well. It can also be made by putting calcium carbide rocks in water. These are also fairly easy to obtain, since they were used in carbide lamps, although pressurising the gas and storing it in a tank might be challenging.
Liquid petroleum gas (LPG) is easily purchasable in some parts of the world, and burning it generates flames near 2500°C.
Kerosene has the advantage of being liquid (therefore, arguably easier to shoot to longer distances or mix with some jellying agent), and generates flames around 2100°C. It's also easy to buy, since it's used as fuel for lamps.
(Temperatures from this Wikipedia article unless otherwise indicated)
It's important to remark that using the flame temperature as base damage might be inaccurate though, since military flamethrowers generate flames around 1200°C, so there might be other factors involved in the choice of fuel other than temperature—gas jelly is used in part for being "sticky", which I assume increases the chances that it will stick to the target.