The only cheap way to carbonate water is in a batch process, like a SoadaStream or equivalent DIY method of releasing CO2 out of a straw into water in a pressurized vessel.

If you really want a faucet that dispenses carbonated water, the cheapest method is to buy just the part of a commercial machine that does the carbonating. It consists of a high-pressure tank with a carbonated water output, and separate water and CO2 tank inputs, as well as a pump that pressurizes the input water, to match the CO2 pressure, so that the water can enter the tank.

If you search Craigslist or eBay for "carbonator with pump" you can usually find something for ~$200. Throw in another ~$100 for a used tank, regulator, and faucet, and you can get a 1920's style seltzer fountain in your own home, then add syrups if you want flavored sodas, or throw in some milk and chocolate syrup and call it an egg cream, despite it containing neither.