While the best way to test for hydrocarbons in drinking water is to get a specialized test kit like the one you posted in your edit or to send a water sample to a lab, there might be a cheaper DIY way to do it. I haven't tested it but I think it is worth a try:
Detecting hydrocarbons is exactly what is typically done when detecting leaks due to broken head gaskets in combustion engines. A broken seal in the combustion chamber of the engine can push some vaporized gasoline into the coolant. Therefore, the standard way to find such leaks is to test for hydrocarbons in the fumes that come out of the coolant when opening the radiator cap. To do so they sell very cheap tests that allow you to suck up those fumes and get them to mix with a chemical reactant which changes color if there are hydrocarbons present.
Here is an example of the test:
And in this video, you can see how it works.
If you Google "head gasket leak test", "combustion leak kit", or "block test" you will find tons of options, and you will very likely also find it also in local shops.
However, the concentrations in your tap water would be much lower than in the design application of these tests. Therefore, the detection that takes minutes in a car might take hours in your tap water. I envision that you may be able to leave the tap open with one of these tests continuously sucking the fumes coming out of the water (if any). To do so, you can put a tube from your vacuum cleaner to one end of the test device (adjusting the size and leakage of the tube so it provide a reasonable amount of suction), so it will get the fumes to bubble through the liquid as long as you want.
This is a very rough hand-drown illustration of the setup I'm describing:
If you get it to change color, then you will then need to figure out the concentration. That would require some additional tests, perhaps doing the same with water with a known amount of hydrocarbons added. Alternatively, you can use that evidence to decide whether to go forward with a professional testing or not.