I have a small tote box setup in my greenhouse that I use
for growing peppers. This is my second year and I've
had good luck with 2 small aquarium air pumps. I don't
use a pump to circulate the water. I change it about once
a month. The air pumps are about seven dollars each.
If you are growing outside you will need to find a way to keep the roots cool. The water level in the box is important also. It should be below your pots. What are you using for nutrients ?

Here is a link that might be useful: Air Pump Link

I have not had a great deal of success growing peppers tomatos and all leafy vegetables are no problems , my peppers get good root growth below but my peppers just seem to sit and sit with no visible extra growth .
I have DWC and air stone and PH and Ec settings correct.
In frustration after being about a foot high I replanted in my soil garden

These are jalapenos , last year I did poblanos. Peppers
like a warm sunny location. It gets very warm in my small
greenhouse. My tomatoes are outside in DWC. I add some epsom salts to both.

Your problems are probably related to the lack of an air stone.

You will want to make sure that you get an air pump big enough to aerate the amount of water you have.

I have provided a link for a single site DWC bucket which has everything you need to start growing, including airstone and pump.

Here is a link that might be useful: DWC bucket system complete

DrKnow thanks in another forum I have mentioned that I am off this morning to get an airstone suitable for my 200 litre reservoir.