I would lean to not having the child in care if they truly can't crawl/hobble about. But, is this really the case? Kids heal quickly and don't really sit still for long...is the child truly going to be immobile and completely dependent?
Obviously there is no way for you to know and if the parents truly believe so they will have to line up alternate care. I suppose I would lay out very VERY clear boundaries for admitting the child into care and leaving it to them to meet those boundaries.
Hopefully by outlining your boundaries they will get the info they need to assess the expected healing time. Parents often over play things related to children so they may be making it seem worse than it is out of parental concern. I ran a girl guide group and one week the mom, in tears, told me a girl wouldn't be back for a while as she was having brain surgery a few days later. I was shocked and sadden and utterly confused when said child showed up next week with dad. When i asked what precautions I needed to make to ensure her safety the dad gave me a weird look and said 'she had tubes put in her ears, don't peg her in the head with a ball and she'll be fine" hahahahah this still cracks me up. This tubes in the ear surgery was common and minor but mom totally saw it as "brain surgery" and was expecting the worst.
I would have them figure things out in their end (by consulting doctors etc) to see if their child can meet your needs! I don't think it is unreasonable to not care for a 3 yr old that needs to be carried everywhere. It's one thing for a parent to do at home, but a whole other thing for a caregiver of multiple needy children to do!