Find out if there is any eco-store or zero waste store nearby. They usually have a wide range of eco-friendly cleaning materials, like ecover.
Sadly, I find that eco-friendly cleaning materials not always work, so occasionally I use something stronger, but try to limit it their use.
I have eczema too, so I know a flare up is terrible. Everyone's case is different, so my experience might not help, but I would focus on the cloths and soaps that touches your kids skin. I also noticed that the water in the area can make a difference, whenever I go back to Hungary or down to London, where the water is harder, I get a flare up (Water in Manchester is really good). Dry, or bad quality air (smoke, dust, mold spores) also makes it worse.
For clothes, use cotton rich (above 95% if possible) soft materials, especially for the layers that touches the skin. Use fragrance free, washing liquid/power/sheet or soap nut and you might find using vinegar or nothing instead of fabric softener better (but the washing machine manufacturers don't like you to use vinegar, so I try to avoid this, but did use it in the past). If the cloth is heavily stained, very smelly or developed that musty smell and the eco-friendly solution is not working, I resort to Aldi's washing powder. It is at least cruelty free, but not good for the environment
Soaps, fragrance and coloring free, or use aqueous cream instead.
I am not a fan of smol or similar, they have really nice fragranced products, but I am not convinced their worthiness.
For general surface cleaning, I mainly use washing up liquid. I have some spray bottles and fill it up with water and squirt some washing up liquid and it is ready to go. If I need more abrasion I sprinkle baking powder on the stain, let it it sit a bit and scour it off. It works well on soap residue in the bath/tiles too.
If you are like most household I have seen in the uk and you have to deal with mold on the grout or silicone (or on the walls), I couldn't find anything else that would work than bleach, but I only use it as a last resort. (Vinegar or steam can ruin your grouts and makes you more prone to mold, plus it is not working that well). If you are lucky, in very early stages, you might be able to scrub the mold off just with the washing up liquid solution. Prevention works well too, we use squeegee on the tiles, ventilate the bathroom and use a dehumidifier in the colder months. I mix a spray solution from general bleach in a spray bottle in 1:1 dilution. (Water first, then bleach)
For window cleaning you can use clear vinegar. Vinegar is also good for descaling kettle, although I find citric acid less smelly.
I buy eco friendly dishwasher tablets, toilet cleaners, and floor cleaners, but only because washing up liquid would be too harsh to use regularly on wood. If I had laminate or similar flooring, I would just use water with a bit of washing liquid, but I heard someone using shampoo(?!)
Shampoo, washing-up liquid, washing liquid and shaving cream is good on fabrics, like sofa or carpet.