Add a class to the HTML <body> tag with React?

With the new React (16.8) this can be solved with hooks:

import {useEffect} from 'react';

const addBodyClass = className => document.body.classList.add(className);
const removeBodyClass = className => document.body.classList.remove(className);

export default function useBodyClass(className) {
    useEffect(
        () => {
            // Set up
            className instanceof Array ? className.map(addBodyClass) : addBodyClass(className);

            // Clean up
            return () => {
                className instanceof Array
                    ? className.map(removeBodyClass)
                    : removeBodyClass(className);
            };
        },
        [className]
    );
}

then, in the component

export const Sidebar = ({position = 'left', children}) => {
    useBodyClass(`page--sidebar-${position}`);
    return (
        <aside className="...">
            {children}
        </aside>
    );
};

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 });
}