That valve will attach to a standard female fitting, which you can put on the end of a hose coming from a pressure regulator, which will attach to a full-size CO2 cylinder available from a brewing or gas supply shop. CO2 refills are a lot cheaper this way.
Put cold water in the bottle with some extra space at the top. Squeeze out the air and attach the valve cap. Set the pressure regulator, connect it to the bottle, open the regulator's output valve, and watch the bottle that was slightly crushed by your squeezing expand back to its normal shape. Slosh the water around with pressure applied for maybe 10-30 seconds. Close the output valve and disconnect.
Voilà. Carbonated water.
IIRC, PETE soda bottles are pressurized to about 50 psi for retail shelves. I don't think they're likely to burst until well beyond 100 psi, and they'll deform before they burst, so if you're careful, you can go a little higher than 50 and make fizzier water than what you can buy in the store. I have used 70 psi many times.
Read up on precautions for handling pressurized gas before doing any of this. Wear eye protection. Don't turn your bottle or gas cylinder into an unguided missile. :)
Sadly, I don't have any info on microplastics released by this process. (Nor by countertop carbonators and their rigid plastic flasks.) I wish I knew of a suitable steel bottle to use instead.