There's nothing illegal about having someone change your diaper or treat you like a child. However, I wouldn't contact agencies in your effort to find a sitter.
I actually tried that tact about 30 years ago; a friend owned a nursing agency. I told her what I wanted and she promised to try to find someone who could help me. After 2 weeks, she told me none of the healthcare providers she had on call wanted the job. I then tried another agency (with an owner I did not know) and got a similar result. The problem is in the unique nature of the job; these folks are paid the same hourly rate no matter what sort of care they provide, and they'd rather not tackle anything strange or unusual. There's just no incentive for them to take any job that isn't considered 'typical.'
Having wasted a month of my time, I turned to my secretary and asked her who she knew in healthcare. Then I sat down at her desk and explained why I wanted to know. Suffice it to say this was about 1988, before the internet had made infantilism somewhat mainstream.
Turned out she wanted to try babysitting me herself. She was probably the worst babysitter I've ever had, but she was a start. Because she worked for me, I instinctively trusted her. Her positive response – and the feedback she gave me – led me to start trying to find people on my own. However, I made up my mind to look for someone with healthcare experience, because even if she wasn't good at roleplaying, she'd at least be competent at bathing, feeding, changing, dressing, etc. That's how I ran into the waitress who was a nursing student.
If you're determined to try the agency approach, good luck! I'm not sure how far you'll get. Patient privacy laws being what they are, I don't imagine you'll have any worse luck than having someone hang up on you. But you'll have far better luck trying to find an individual, having an honest conversation with him or her, and offering them a decent hourly rate to try caring for you. My goal was always to help a new sitter have so much fun the first time that she wanted to come back again and again. That's only possible if you have a personal relationship with the caregiver and show genuine appreciation for what she provides.
I hope that helps.