Possible in IE and chrome although it does not work in firefox:

<script>
   function toggleZoomScreen() {
       document.body.style.zoom = "80%";
   } 
</script>

<img src="example.jpg" alt="example" onclick="toggleZoomScreen()">

Try if this works for you. This works on FF, IE8+ and chrome. The else part applies for non-firefox browsers. Though this gives you a zoom effect, it does not actually modify the zoom value at browser level.

    var currFFZoom = 1;
    var currIEZoom = 100;

    $('#plusBtn').on('click',function(){
        if ($.browser.mozilla){
            var step = 0.02;
            currFFZoom += step; 
            $('body').css('MozTransform','scale(' + currFFZoom + ')');
        } else {
            var step = 2;
            currIEZoom += step;
            $('body').css('zoom', ' ' + currIEZoom + '%');
        }
    });

    $('#minusBtn').on('click',function(){
        if ($.browser.mozilla){
            var step = 0.02;
            currFFZoom -= step;                 
            $('body').css('MozTransform','scale(' + currFFZoom + ')');

        } else {
            var step = 2;
            currIEZoom -= step;
            $('body').css('zoom', ' ' + currIEZoom + '%');
        }
    });

You can use the CSS3 zoom function, but I have not tested it yet with jQuery. Will try now and let you know. UPDATE: tested it, works but it's fun

I could't find a way to change the actual browser zoom level, but you can get pretty close with CSS transform: scale(). Here is my solution based on JavaScript and jQuery:

<!-- Trigger -->
<ul id="zoom_triggers">
    <li><a id="zoom_in">zoom in</a></li>
    <li><a id="zoom_out">zoom out</a></li>
    <li><a id="zoom_reset">reset zoom</a></li>
</ul>

<script>
    jQuery(document).ready(function($)
    {
        // Set initial zoom level
        var zoom_level=100;

        // Click events
        $('#zoom_in').click(function() { zoom_page(10, $(this)) });
        $('#zoom_out').click(function() { zoom_page(-10, $(this)) });
        $('#zoom_reset').click(function() { zoom_page(0, $(this)) });

        // Zoom function
        function zoom_page(step, trigger)
        {
            // Zoom just to steps in or out
            if(zoom_level>=120 && step>0 || zoom_level<=80 && step<0) return;

            // Set / reset zoom
            if(step==0) zoom_level=100;
            else zoom_level=zoom_level+step;

            // Set page zoom via CSS
            $('body').css({
                transform: 'scale('+(zoom_level/100)+')', // set zoom
                transformOrigin: '50% 0' // set transform scale base
            });

            // Adjust page to zoom width
            if(zoom_level>100) $('body').css({ width: (zoom_level*1.2)+'%' });
            else $('body').css({ width: '100%' });

            // Activate / deaktivate trigger (use CSS to make them look different)
            if(zoom_level>=120 || zoom_level<=80) trigger.addClass('disabled');
            else trigger.parents('ul').find('.disabled').removeClass('disabled');
            if(zoom_level!=100) $('#zoom_reset').removeClass('disabled');
            else $('#zoom_reset').addClass('disabled');
        }
    });
</script>

as the the accepted answer mentioned, you can enlarge the fontSize css attribute of the element in DOM one by one, the following code for your reference.

 <script>
    var factor = 1.2;
    var all = document.getElementsByTagName("*");
    for (var i=0, max=all.length; i < max; i++) {
        var style = window.getComputedStyle(all[i]);
        var fontSize = style.getPropertyValue('font-size');

        if(fontSize){
            all[i].style.fontSize=(parseFloat(fontSize)*factor)+"px";
        }
        if(all[i].nodeName === "IMG"){
            var width=style.getPropertyValue('width');
            var height=style.getPropertyValue('height');
            all[i].style.height = (parseFloat(height)*factor)+"px";
            all[i].style.width = (parseFloat(width)*factor)+"px";
        }
    }
</script>

Easiest Solution

To get window zoom (not CSS zoom), there are some ways, but the easiest of them is as follows:

(function() {
    window.addEventListener('resize', evt => {
        let zoom = (window.outerWidth / window.innerWidth).toFixed(2)
        console.log('zoom: ', zoom, '%')
    })
})();

Another Way

An alternative way is window.devicePixelRatio, but it may display incorrect value, depending on screen resolution (specially with multiple screens).

browser.tabs.setZoom() should handle this case.

let zooming = browser.tabs.setZoom(
  tabId,           // optional integer
  zoomFactor       // number
)

See the MDN Docs: https://developer.mozilla.org/en-US/docs/Mozilla/Add-ons/WebExtensions/API/tabs/setZoom

<html>
  <head>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
        <script>
        var currFFZoom = 1;
        var currIEZoom = 100;

        function plus(){
            //alert('sad');
                var step = 0.02;
                currFFZoom += step;
                $('body').css('MozTransform','scale(' + currFFZoom + ')');
                var stepie = 2;
                currIEZoom += stepie;
                $('body').css('zoom', ' ' + currIEZoom + '%');

        };
        function minus(){
            //alert('sad');
                var step = 0.02;
                currFFZoom -= step;
                $('body').css('MozTransform','scale(' + currFFZoom + ')');
                var stepie = 2;
                currIEZoom -= stepie;
                $('body').css('zoom', ' ' + currIEZoom + '%');
        };
    </script>
    </head>
<body>
<!--zoom controls-->
                        <a id="minusBtn" onclick="minus()">------</a>
                        <a id="plusBtn" onclick="plus()">++++++</a>
  </body>
</html>

in Firefox will not change the zoom only change scale!!!

You can use Window.devicePixelRatio and Window.matchMedia()

const onChange = e => {
  const pr = window.devicePixelRatio;
  const media = `(resolution: ${pr}dppx)`;
  const mql = matchMedia(media);

  const prString = (pr * 100).toFixed(0);
  const textContent = `${prString}% (${pr.toFixed(2)})`;

  console.log(textContent);
  document.getElementById('out').append(
    Object.assign(document.createElement('li'), {textContent})
  )

  mql.addEventListener('change', onChange, {once: true});
};

document.getElementById('checkZoom').addEventListener('click', e => onChange());

onChange();

<button id="checkZoom">get Zoom</button>
<ul id="out"></ul>