You dont have to use chemicals, as long as you clean them it will reduce the chance of illnesses. I dont even use chemicals, just an abrasive sponge and scrub the waterers under water. But i have rainwater and natural ponds, but the seven waterers that I use I keep clean and scrub every other day at least. I noticed that having cleaner water reduces problems.
Bleach or iodine would work too. Vinegar isnt really a strong disinfectant, but more important is just using an abrasive sponge and water.
How is the eye swelling doing?