By default, the input type is defined as a button:
<input type="button" value="{$chat.supportrequest.openchat$}" />
For example, if you wanted to change this value to an image, you also need to define the onclick value parameter:
<input type="image" onclick="return false;" value="{$chat.supportrequest.openchat$}" />
Otherwise, you get the error message:
The server method GetSupportEngineersOnlineCount failed.