That's a heck of a system. Yes, you can open that well, there are bolts on the top of the seal, where the two smaller pipes enter the white pipe, but it would be very hard to do. Not worth it. The bolts connect 2 steel (or cast iron) plates with rubber in the middle. By tightening the bolts, the rubber is squished out, making a good seal.
One way you could add chlorine is at the filter. unscrew the filter, dump out the water, and fill the filter housing with bleach. run the water for a few seconds, and repeat. this will fill your tank, then run it to all the faucets, like discussed previously.
Oh, you really only need to fill the filter housing twice, it should be less than a gallon of bleach.