We use White Vinegar a lot, including for laundry instead of softener. It might be a good idea to wear gloves when you use it for cleaning if you have dermatitis and just wet cloth it away off surfaces afterwards.
A Canadian brand called 'Attitude' does a good unscented laundry detergent. They're available here in the U.K so should be in the U.S. They do a whole range of household products too I think. Bio-D is a U.K brand which may or may not be available where you are. Their fragrance Free Sanitising hand wash is brilliant for cleaning everything from your hands/hair to the floor! Vodka is good for if you want a fragrance free sort of semi sterilizing thing - either for skin wounds/spots or even to spray down items if needed. Not the cheapest option but it works! It Works! blocks - available on QVC and probably Amazon etc - they're used with just water and are fantastic for cleaning surfaces / scuff marks / sinks etc etc etc - no chemicals no smells. Just go easy with them on painted/glossy surfaces as they can slightly remove the top layer sometimes. (There's a video on the qvc website for how they work). Microfibre cloths/ floor mops - microfibre cloths are fab. Use with just water or with whatever cleaning solution you want to put on them to enhance the efficacy. 'Catch Mop' is quite good. E-Cloth - this is a very thin cloth that you use either dry or with just water depending on where you're using it - works fantastically on mirrors/windows/t.v screens computer screens etc..
Agree with the Method floor stuff as Delta said above. I can't take scents at all, but this is just very barely there lemony. Bioderma Sensibio H2O Make-up Removing Micelle Solution - yes, I know, it's make-up remover, but, all past using ability of chemicals included, this stuff is hands down THE best stain remover for fabrics (even delicates) / carpets we've ever come across! And I'm talking it gets out Sharpie pen/ tomato sauce, the lot - if it doesn't do it on the initial douse and rub, the trick is to douse and douse again, this stuff keeps working while it's on there. I don't have a clue how it does it but it does, and it doesn't even sting when you get it in your eye taking mascara off! The stuff is amazing! And of course, it doesn't have any fragrance to it either. Johnson's Baby Extra Sensitive Wipes again, extremely gentle on skin, no scent, but wow do they work for cleaning the house!
A bit of cooking oil will help get sticky marks/label residue off, then soap or vinegar or baby wipe the oil away.
Also, things like toothpaste clean really well. (I used it to clean the bathroom sink when I was a little girl - mum was horrified Cut to years later and I actually found the tip in a cleaning book called Queen of Clean! Obviously not for large areas, but As they say - Necessity is the mother of all invention - mmmm minty fresh
