Hi there ... welcome to the board. :O)
I really like the Super SensitiveShampoo & Bodywash by California Baby for sensitive kids' skin. It's not great as a shampoo unless you don't mind lathering up twice, but it makes a great skin wash. No tears. I think you can order a pump to fit the bottle, otherwise it's a pop-top that you pour and it is a bit watery, so your kids might use too much (mine do). I would recommend the entire California Baby Super Sensitive line. That lotion is one of very few I can use on my skin without irritating my acne and very sensitive face, and it has also been one of even fewer successful lotions I can use on by 'eczema & hives' boy without irritation. Same for the CB fragrance free sunscreens ..... LOVE them!!
I also like the Aquaphor (Eucerin) Gentle Wash & Shampoo. It's very gentle and is actually 'tear free'. I know many kids' products claim to be tear free but burn anyway ... not this one. It's soap free, fragrance free, dye free & non-irritating (per the label) and I agree 100%. The only option it comes in is a pop-top pour, so there is the possibility of kids wasting it, but it's great stuff.
My boys LOVE the foaming Johnson's baby wash. I know you specifically requested no fragrance, so I hesitate recommending this one, but it doesn't bother my most sensitive-skinned little guy and it's completely kid friendly in use. It is soap free, dye free, and hypoallergenic. Since it foams, they can pump it and get a handful of bubbly foam to use all by themselves, very little mess and no wasting (except the times I catch them pumping it for fun and extra bubbles, LOL). Might be worth a try.
If these don't help, let me know and I'll dig up some more. These are just my personal favorites. Good luck! :O)

Thank you Steph for the great siggy!

