I don't know of any web site that you can use to check back grounds that would be complete enough for this purpose. There are sites to be found that will do background records checks on people. You do have to pay. I've seen the sites when I've been looking to locate a friend with whom I've lost touch. Google background checks, criminal record checks, that sort of thing.
If you go thru one, I'm not sure that you can trust that the info is even accurate. For fun my cousin used one to find out what was available on line for her and her husband. She said not one fact was true. She's white and it said she was black. That's just he start.
Our state, Oregon, certifies evening and night time child caretakers. There may not be anyone that has asked to be certified but if you find someone then you can ask them to get certified.
I suggest that you ask the applicant for references and clearance for you to do a criminal background check. Call your local police department records section and ask what they need to do a background check for you on a prospective caretaker. You may need to take the person with you to the records section but it's more likely that you'll need permission from them signed in front of a notary public along with certified copy of their i.d. I'm not sure how to get that but someone in the police department will know. I'm not sure that this is done often enough for any clerk to know the answer; so don't accept the first answer.
Talking directly with people who have had this person care for their children will give you more information and a better idea of the sort of person your prospective caretaker is. It's often thought that the people that one gives as references will only give good information. This is just not so. It's surprising how often people are really unaware of how they are seen by others.
Sometimes you can get information on a person by googling their name. There should also be a state website that lists sex offenders.
When you interview, ask lots of questions and pay attention to facial expressions and body language as well as answers.
I also suggest considering hiring a high school girl to come in for evenings. When I was in high school I worked for a family who owned an all night restaurant. I frequently slept at their house and spent some weekends there.