Add a class to the HTML tag with React?

TL;DR use document.body.classList.add and document.body.classList.remove

I would have two functions that toggle a piece of state to show/hide the modal within your outer component.

Inside these functions I would use the document.body.classList.add and document.body.classList.remove methods to manipulate the body class dependant on the modal’s state like below:

openModal = (event) => {
  document.body.classList.add('modal-open');
  this.setState({ showModal: true });
}
hideModal = (event) => {
  document.body.classList.remove('modal-open');
  this.setState({ showModal: false });
}