In the content.js of your Chrome Extension, you'll want to create and append your modal element to the body of the page on load. You can do that with something like:

    const modal = document.createElement('h1');
    modal.innerText = 'This is the "modal"';
    modal.style.visibility = 'hidden';
    document.body.appendChild(modal);

Then add event listener to the button using CSS selectors and addEventListener

const button = document.querySelector('button.large-button-primary');
button.addEventListener('click', showModal);

And a function which changes your modal's visibility once it's clicked

const showModal = () => {
   modal.style.visibility = 'visible';
};

<button _ngcontent-fes-c25="" class="large-button-primary width marTop16">
    <span _ngcontent-fes-c25="">
        Continue
        <!----><!----><img
            _ngcontent-fes-c25=""
            alt="disabled next icon"
            class="ng-star-inserted"
            src="https://static.incometax.gov.in/iec/foservices/assets/buttonIcons/nextIconDisabled.svg"
        /><!----><!----></span
    >
</button>

<script>
    const button = document.querySelector('button.large-button-primary');
    const modal = document.createElement('h1');
    modal.innerText = 'This is the "modal"';
    modal.style.visibility = 'hidden';
    document.body.appendChild(modal);

    const showModal = () => {
        modal.style.visibility = 'visible';
    };

    button.addEventListener('click', showModal);
</script>

If the event listener isn't working, then that's probably because you're setting the button variable before the target button is even on screen, which means the querySelector is just returning undefined.