You can create a js function and then call it with the anchor to change the value attribute of the input. Input type text should not have a closing tag either.
function changeChatInput(value) {
document.getElementById('chatInput').value = value;
}
<form method="POST" action="chat.php">
<input type="text" name="message" id="chatInput" />
<input type="submit" />
</form>
<a href="javascript: changeChatInput(':cat');">
<img src="cat.jpg" />
</a>
</div>