Do states have the power to make school uniforms in public schools mandatory?

Twenty-two states authorize schools and districts to implement dress code and/or uniform policies. ... In fact, court rulings and attorney general opinions in an additional four states have upheld the right of districts and schools to set dress requirements. No state requires students to wear uniforms.

They could, yes. But most of them haven't, because it's recognized that not all students have families that can afford to buy uniforms. Most schools that require uniforms are usually private or parochial schools, not public schools.

Even local schools boards and schools themselves have the ability to do this, as they set their own dress codes. 

Yes, states have the power to make school uniforms in public schools mandatory. However, this is never going to work. 

My sister went to a private, all-girls high school in Pennsylvania, despite living in South Jersey. She had a dress code, which required her to wear either her spring/summer uniform (during warm weather) or fall/winter uniform (during cold weather). She also had to wear a formal pair of shoes, although she was allowed to wear sneakers during the spring. However, during the summer after her junior year of high school/before her senior year of high school, she bought a nice pair of size 8 Adidas Falcon W cleats. Of course, during her senior year of high school, she changed from her school shoes to her sneakers once her school day was over. (Her bus came 3 hours after her school day was finished.)

Of course they do.

When any corrupt institution, i.e. the government, wants to implement something they will.