How to replace an HTML element's text with JavaScript when the element doesn't have a class?

What about selecting it by using its DOM order?

var parent = document.querySelector("select");
var element = parent.querySelectorAll("option")[1];
element.innerHTML = "watch the webinar now";

This article can help: How to get the child element of a parent using JavaScript?

You can use querySelector to get an specific child inside an element and change it.

Snippet

In this case I use querySelector(':nth-child(2)') to get the second child.

var parent = document.querySelector(".someClassName");

console.log(
  parent.querySelector(':nth-child(2)').text
)

document.getElementById('btn').addEventListener('click', () => {
  parent.querySelector(':nth-child(2)').text = 'asdasd asdasd';
})

<button id='btn'>change</button>

<select class="someClassName">
  <option>aker including versions of Lorem Ipsum</option>
  <option>Donec semper tortor ac velit tempus,</option>
  <option>vel pellentesque lorem gravida</option>
</select>

To change the text of the second option

document.getElementsByTagName('select')[0].options[1].innerHTML = "watch the webinar now";

If the option is not always second, you can give the option a id then use that in your query selector.