It would not be an issue to publish the template to your website but probably simpler to use SharePoint.
Link to the template.xsn in SharePoint to open a new form or a link to the xxxx.xml file in the library and windows will open that xml document in InfoPath on the client's machine.
If you need a way to display available files, you could embed the SharePoint library in a frame or use the list webservice, like you would in InfoPath, to display the files in the library and give them a link to each.