Is it ok to let seniors crawl?

No. It is not okay to let an elder crawl around on the floor like an insect because they are in pain or for fear of falling.
Your MIL cannot be left living alone in her house anymore.
Sorry, but your SIL is not her primary caregiver. If your MIL had one of those she wouldn't be crawling around her house to get from one area to another.
Your SIL probably does think her mother crawling around the house alone is fine because it means mom's staying out of a care facility hence preserving future potential inheritance. Or because the crawling keeps mom from having to possibly be moved into her house.
I think you would do well to not worry about potentially damaging family relationships by reporting to APS what's going on here. Your worry would be better placed in some thought about what happens if there's a fire in the house and your MIL can't crawl to an exit to get out. I'd worry more about that.
You know, I've been a homecare caregiver for almost 25 years. I have been in every situation that can possibly happen in a person's home. I have seen and heard it all. I've been on this forum for a long time too and have seen every manner of story here.
I can't believe that someone, a teacher, would ask such a question as "Is it okay to let the elderly crawl?" I am truly shocked and that is no easy thing to accomplish.
So, I will answer your question in the plainest possible language.

NO! IT IS NOT FREAKING OKAY TO LET THE ELDERLY CRAWL AROUND ON THE FLOOR LIKE INSECTS TO GET FROM ROOM TO ROOM!

IT IS NOT OKAY FOR AN ELDERLY PERSON WITH DEMENTIA WHO CRAWLS TO GET AROUND TO BE LIVING ALONE UNSUPERVISED!

None of this is okay.

Keep your nose out of it, do not make your SIL life harder, she already has a lot on her plate. Last thing she needs if for people to get involved and have them interject their opinions and solutions that may result in your MIL falling and potentially breaking something then leaving it up to the sister to clean up the mess. I highly doubt you guys take much of your time to help, as you said you are working and busy.

Busy bodies are the worse, mind your life and let other mind theirs. Unless some abuse is taking place do not make your SIL life harder by getting APS involved. Caregiving is hard enough as is, no need to do it with a agency looking over your shoulder, having to do month checks and what have you. Please.

Crawling seems unnecessary if she was in the right setting. Sounds like she needs, at minimum, assisting living or a nursing home. In 27 yrs of clinical work in a nursing home I never saw a resident crawl around. They were put in wheelchairs with assist. Sounds very sad to me.

worrieddil- Why report to anyone? If you think you can do better at taking care of MIL, then take over the care. I'm sure SIL would love to have a big long break from the stressful caregiving.

I am a teacher too and mandated reporting is for when you suspect abuse or extreme negligence. I don't think this falls into either of those categories. I would, as others have suggested, have a family meeting to brainstorm what would be the best way to help her. Then you aren't stepping on toes and creating a rift that won't help anyone.

elderly crawling is dangerous as their skin is paper thin and prone to tearing.

Its easy on the outside to sit and criticize. Instead of feeling like it's necessary to file a report why don't you offer to go spend the night with your mother in law a few night a week. If you can't do that then maybe set up a home care aid to stay with her at night. I'm curious to know why you didn't go with your husband to check on her? Whether you like it or not, your mother in law is your husbands family too and you should be offering better solutions not trying to make things more complicated.

Can your Mil walk if she needs to?
Can the back pain be treated?

There are exercises and treatment programs online that recommend crawling as beneficial, and state that crawling can rewire the brain. Maybe that is what is happening?

So maybe, don't panic. Be part of the solution by getting more information and talking to your husband.

I had terrible sciatic pain when I was pregnant with each of my kids. I actually planted a new lawn with plugs of grass when I was eight months pregnant with one of them, and it was the only time I wasn't in pain because I was on all fours with no pressure on my back.

I would suggest to Sis that there might actually be a remedy for Mom, but you won't know unless you get her to the doctor. She might benefit from a Cortizone shot, or some painkillers or physical therapy. You won’t know until you have her evaluated. I would really question Sis as to why she’s so adamant about not getting Mom any help. Back pain is excruciating, and even if her problem is spinal stenosis and nothing can be done, at least you'd know.

I would hope reporting it to APS would be a last resort, and a conversation with your sister would be the more effective route to take.

Is she living alone?
If so with the cognitive decline that might not be a good idea.
Does she have a walker that is sized for her and does she know how to PROPERLY use it?
Maybe an assessment by a Physical or Occupational Therapist might be a good idea.
It actually sounds like she needs more help, assistance than she or the rest of the family wants to admit.