You need to use the Web Storage API. As this seems like a plugin you're building, I would wrap the Storage API methods and use a prefix to prevent interference with other scripts that may be running on the page (as I have done in the example below). 
I've taken the liberty of expanding your example a bit to better demonstrate usage, and I've removed all of that jQuery nonsense.
Here's a demo showing it in action
const textResizePlugin = scope => {
    if(!(scope instanceof Node)) {
        throw new Error('Invalid scope supplied');
    }
    const style = scope.style;
    const prefix = 'textResizePlugin_';
    if(scope.classList.contains(`${prefix}active`)) {
        throw new Error('Already initialized');
    } else {
        scope.classList.add(`${prefix}active`);
    }
    const storage = {
        get: (key) => window.localStorage.getItem(prefix + key),
        set: (key, val) => window.localStorage.setItem(prefix + key, val)
    };
    const setSize = size => {
        storage.set('fontSize', size);
        style.fontSize = size;
    };
    setSize(storage.get('fontSize'));
    const resize = m => setSize(
        (style.fontSize||"1.0em").replace(/[\d.]+/, num => +num + m * 0.2)
    );
    const html = `
        <div id="${prefix}bar">
            <button id="${prefix}less">-</button>
            <button id="${prefix}more">+</button>
        </div>
        <style type="text/css">
            #${prefix}bar {
                background: rgba(200,200,200,0.3);
                position: fixed;
                font-size: 0px !important;
                top: 0;
                width: 100%;
                padding: 5px;
                text-align: right;
                box-sizing: border-box;
            }
        </style>
    `;
    scope.insertAdjacentHTML('afterbegin', html);
    const [less, more] = scope.querySelectorAll(`#${prefix}less, #${prefix}more`);
    less.addEventListener('click', () => resize(-1));
    more.addEventListener('click', () => resize(1));
};
textResizePlugin(document.body);