Another vote for charedim. By far the thinnest and I got 2 pounds of pupa Salem last year to compare.
You can just walk into the actual bakery and they will allow you and your kids to watch everything from the mixing, kneading to the actual baking. No touching anything though.
In regards to actual buying matzo at the bakery, from purim till pesach every day is baked by a chabura. But if there is any extra they will sell it to you. The closer you are to pesach the less likely they will have extra.
There is also a children's matzo bakery on Ocean parkway by Brighton Beach in a chabad shul/school if I remember correctly.
Enjoy!