Is it possible to convince my 95 yo MIL to let someone shampoo her hair?

MOre details. She has refused about all help in this bathing/hair care manner. We got her a shower chair after the first time she got stuck in the tub two years ago and had to be taken to the hospital because she passed out after granddaughter fished her out. She moved it out of the tub and refused to use it. We had her evaluated by the nurse for the county council on aging and she was EXTREMELY uncooperative and wanted nothing to do with her. She says she hates showers and only takes baths. Got her the wand type shower head for the bath chair to make rinsing easier. She somehow managed to climb up there and take it down. She is too short to reach without something to stand on. I put it back up and it stayed. We offered to get her a walk in tub. Nope. She got stuck in the tub for the second time in January. That was when I brought back the "bath chair" with the threat that next time the squad would have to fish her out butt naked if she got stuck. Nope. She moved it out again. I have offered to shampoo her hair at the sink. Nope. Asked her how she washes her hair. She says leaning over the tub. Said I would help her do it that way. Nope. Offered to take her to the salon for at least two years. Nope. I am so tired of Nope. Why is being stubborn what she remembers best? You can imagine how well she would do in assited living. Nope. Nope. Nope. They would HATE her. My daughter did the dry shampoo. Grandkids are about the only bribe that works and there are no cute little great grands. All three of our adult children are childless. :( We requested that she do respite care while we were on vacation and you can guess how that went. She is so close to deaf, I am sure her neighbors down the street can hear when I have to yell to try to talk to her. Won't wear her hearing aids. I just give up and go home. I think that is what she wants. Husband is only child and ONLY LIVING relative. He hates dealing with her.

My mom was part of the weekly wash and set generation, do you think it might work to take her to a salon?

Your mil is 95 and suffers from Alzheimer's/dementia according to your profile, meaning she's not being uncooperative on purpose. Of course you're not being too picky to expect a shower and hair shampoo about once a week, which is reasonable. Hire someone to shower her 1x a week or else place her in Memory Care Assisted Living, her choice. But first figure out what she's afraid of in the shower. For my mother it was the slippery shower floor so I bought her water shoes on Amazon which solved the problem. For some it's the shower head water spray they hate, which a hand held unit helps with. Others get cold and need the room warmed up. Teepa Snow has good videos on YouTube about techniques to use to coax dementia patients into bathing. Check them out.

I suggest you read this 33 page booklet online about managing dementia and what to expect with an elder who's been diagnosed with it. Showers are discussed in the booklet as well as lots of Do's and Dont' s for dealing with dementia sufferers.

Understanding the Dementia Experience, by Jennifer Ghent-Fuller 

https://www.smashwords.com/books/view/210580

Jennifer is a nurse who worked for many years as an educator and counsellor for people with dementia and their families, as well as others in caring roles. She addresses the emotional and grief issues in the contexts in which they arise for families living with dementia. The reviews for her books are phenomenal b/c they are written in plain English & very easy to read/understand. Her writings have been VERY helpful for me.

The full copy of her book is available here:

https://www.amazon.com/Thoughtful-Dementia-Care-Understanding-Experience/dp/B09WN439CC/ref=sr_1_2?crid=2E7WWE9X5UFXR&keywords=jennifer+ghent+fuller+books&qid=1657468364&sprefix=jennifer+ghent%2Caps%2C631&sr=8-2

Best of luck to you.  

Yes to everything Catskie62 listed. The waterless shampoo and conditioner caps work really well. You just put them in the microwave for 10-15 seconds and then put them on her head, rub it all around, take it off and dry her hair.
You can also buy extra large body wipes so she can clean herself(or you can)without getting in the shower.
Those are great for the in between showers, and both the wipes and shampoo caps can be ordered on either Walmart.com or Amazon.

Is MIL doing the dry shampoo herself or are you helping? Offer to assist her with her shower and then shampoo her hair. Go about it in a non argumentative way. You could always take her to a salon for a wash &style. They also sell no rinse shampoo caps on Amazon.