State of a HTML Checkbox as a Parameter to a new Website

Could the link be to any website or is it just linking between pages on the same site/origin? If you keep it to the same origin you could also use localStorage to hold the state instead of passing it in the query string/parameter.

You could use an event listener to detect when the input has been clicked, then modify the link url.

const checkbox = document.getElementById('crocodiles');
const webpage = document.getElementById('google');
checkbox.addEventListener('click', function () {
  if (checkbox.checked === true) {
    google.setAttribute('href','https://google.com/?q=crocodiles');
  } else {
    google.setAttribute('href','https://google.com/');
  }
});

<label for="crocodiles">Search about crocodiles?</label>
<input type="checkbox" id="crocodiles" />
<br />
<a href="https://google.com/" id="google" target="_blank">Go to Google</a>

Just FYI, you could do this without any JavaScript, by simply using a form:

<form action="/action.php">
<input type="checkbox" name="state" value="checked">
<button>newSite</button>
</form>

I don’t want to have an extra submit button

This would not need an extra submit button, but just a submit button. And if desired, you could still format that submit button to look like your other links do.