The condensate from natural gas burning comes mainly from the burning. Methane is the primary component of gas, and it is chemically CH4. When burned, you have:
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CH4 + 2 O2 --> CO2 + 2 H2O
The relevant parts here are that 2 "oxygen" molecules from the air are consumed, and (after some chemical shenanigans) 2 water molecules are produced.
This is not simple, since natural gas consists of other things than methane, and the reaction is not a single-step reaction, so carbon monoxide and other things are also by-products. But the majority of the gas is methane, and the majority of the results are dioxide and water.
The water, naturally, is a vapor since it's created in the middle of a burning flame. But the high-efficiency gas appliances really try to capture the waste heat and feed it back into the system, so as a result the water vapor is cooled to "condensate". The condensate is not pure water (a lot of those impurities and by-products mentioned above also condense) and so you should not drink it (seriously, you could die), or even allow it to run through your sewer drain (seriously, it will eat a hole in your poo pipes, and nobody wants an that) without neutralizing it.