Yes, you can automate Word using interop and presumably print the docx to pdf that way. But MSDN explicitly says Office Interop can be naughty in terms of potentially breaking your web server scalability. I would trust its words and avoid it by finding a third party doc to pdf renderer - though I know many whom have not.
Be sure that you understand all the considerations for server-side Automation of Office.
Just by googling you should find many doc to pdf conversion libraries - inded there may even be a remote webservice you could call!