Вот попытка, но я бы сказал, что использовать jq намного проще
document.querySelectorAll(".dropdown-menu a").forEach(a=>{
// Assign a click to each element
a.addEventListener("click", function(){
// get the closest eg (parents) and then seacrh for the button
// when you wrote text i assumed you ment innerhtml.
a.closest(".dropdown").querySelectorAll(".btn")[0].innerHTML = a.innerHTML;
})
})
<div class="dropdown">
<button class="btn btn-default dropdown-toggle" type="button" id="dropdownMenu1" data-toggle="dropdown" aria-haspopup="true" aria-expanded="true">
English
<span class="caret"></span>
</button>
<ul class="dropdown-menu" aria-labelledby="dropdownMenu1">
<li><a href="#" data-value="english">EN</a></li>
<li><a href="#" data-value="thai">TH</a></li>
</ul>
</div>