I think you'd have to go to your state child care site and download the state child care regulations booklet but in Oklahoma I don't think they can.
If a child is different, say....they can't sit still for more than 5 minutes and they can't follow you when you tell them something or ask them something, they don't like to be touched or their space invaded, etc...a child care worker who had been working in the field for several years is going to be able to say "hey, there's something different about this child" but they cannot say "Hey, you're kid is autistic".
A child care worker can talk to the director to see if they can find a way for the local psychologist to come in and observe the class to give that teacher some pointers on how she can have a better learning environment, how she can develop a better curriculum for her age group, the teacher can talk to the psychologist about any stresses she's/he's having and how the psychologist can help them feel better.
A child care center can't really call someone and say they're having a problem with a particular child and they need to be referred. That's the parents job.
However, if this child is obviously struggling with some developmental issues that are impairing them from learning or growing and developing the child care staff have a legal and moral obligation to turn the family in to child welfare.
Child care workers aren't like a school employee. They don't typically have BA's in child development or some other related field. They usually have hours and hours and hours in child development and other topics required for them to work in this field. They have specific training required by the state.
So if you are a worker in a child care setting and you have a child that you suspect has some sort of developmental delay or a disability please talk to the director before you even talk to the parents. They're going to take it wrong and you don't want them upset with you. Let them be upset with the director. That way they won't take it out on you. They'll turn to you and ask what's been going on, why haven't you said anything, etc....then you can say "Well, I have seen some stuff I didn't know about so I asked the director if she could help me figure out how to deal with "little Johnny's" actions when someone tried to touch him. I didn't know what I might be doing wrong and I felt I needed a little guidance".
This will hopefully diffuse the situation some and they'll be more receptive to getting "little Johnny" evaluated.
If you're a parent this has happened to then I suggest you ask the director what's going on, why didn't they talk to you first, how they came to the conclusion "little Johnny" needed a referral. Hopefully you'll be receptive and respectful of their years of experience and consider their referral to be something for your child's growth and development.
Head Start/Early Head Start and other pre-school programs that are under the umbrella of the public school system follow different guidelines and can actually ask the school psychologist to come and observe a specific child. They have that right due to being a public school sub program.