When you say "fogger" do you mean the small type for terrariums? Any large reptile supplier should carry them. However, not all of them are very durable. You can create fog for a cham setup using a modified ultrasonic room humidifier (sometimes called a cool mist humidifier). You want to look for a model that has a removable spout where the fog comes out, and attach some flexible tubing or pvc pipe to the opening and direct the fog into your cham's cage. There are many forum threads about doing this. Do a search on humidifiers to see ideas and brands members have used.
Yes, you can use them with a screen cage. Just be aware that some droplets will end up outside the cage, so protect the wall or furniture that might get wet. A screen cage will dry out fast depending on how dry the room is, but you can hold the moisture in longer by adding sheet plastic to the sides or back of your screen cage.