Have you looked at the Solr-driven reserves search? Instructions on how to turn it on can be found above the [Reserves] section of web/conf/config.ini. If you use that module instead of the default version, you get more powerful searching, and it’s easier to create an external form that submits the necessary parameters into VuFind.
If you’re happier with the default drop-down approach, though, one option would be to make a server-side call to VuFind from whatever CMS you are using to build your website – from there, you could extract the necessary form elements. Perhaps you could do this once a day and cache the results to reduce load on your ILS.