It would help to know how old your child is. But essentially if you put into writing a request for an evaluation for special education services to your school principal (in the book, "The ADD/ ADHD Answer book," by Susan Ashley. p 82) They must respond to your request. If you include the teachers and pediatrician's input it would be very hard to deny the request. They must respond (in California) within 15 calendar days.
So yes, definitely, the school can do the testing (at no cost to you) if they so choose. You might want to get the book i mentioned above for other very helpful information. Best wishes.
It is a normal concern, check the board of education, depending on the state law's, usually the school system should be able to provide the test. They usually check after age 3. Also keep in mind some kids around this age does act hyper and that is normal. You just need to find an outlet. Another thing to consider is food allergies, some foods have a tendency to trigger certain behaviors. You might want to watch what your child is eating before certain behaviors.