I'm really unsure now. I just used your exact code on our demo site, and the Apply button shows fine. Feel free to check it out on the page below:
www.sourcecoast.com/jlinked/social-modules
Scroll down to the very bottom for the "Test Apply" button. The only change I made was to add our admin render key.. {JLinkedApply ... key=xxxx} If you don't have the proper key, the actual tag will show. So, that's likely not the issue for you since nothing is showing.
Is there anything you're using to dynamically load content, like through AJAX or something? Can you try switching your template to one of the built-in Joomla templates to see if that explains the behavior? Something seems to be removing our script tags, which is obviously a problem. We just need to figure out what it is.
Finally, would it be possible to access the admin area of your site? If so, please Private Message me the credentials. You can PM me by clicking the arrow under my picture (<----) and hitting the mail icon. That will help understand things more, but isn't absolutely necessary yet.
Thanks,
Alex