When you say you have a plugin for licence checking, my understand it is AutoCAD DLL (.NET, possibly) running inside AutoCAD. Then there is web application that is hosted somewhere that does the actual licence info validation based on the request sent to it and returns the validation result.
In this scenario, the plugin on the AutoCAD side is to gather licence information from user's computer, and then send to the remote web application (most likely, as Http request) and wait the web app on the remote web server responds back (as Http response).
It looks like when you say plugin, it is actually 2 different applications: a) AutoCAD plugin that sends licence info to b) a web app (service) to validate the licence.
Regardless where the web app is hosted, if it is accessed as most web application/website, then your AutoCAD plugin only need to send proper Http request. If the web app allows anonymous access like most public website, then no need to authentication credential. Otherwise you need to provide login information. Since it is hosted on the net, unless your office network is automatically connected with the hosting network via VPN as the same network domain, it is unlikely that Windows authentication can be used. It is really depends on the web application.
Obtaining MS Azure may not help you, if the web application is done by someone else, and only helps you if you are to develop the web application and want to host it with Azure. Still, your plugin need to pass login credential with it web request sent to the web application.
With System.Web/Net namespace, you can make call to remote web application (web services) directly from desktop app (your AutoCAD plugin); with AutoCAD2015, you may even want to try to use JavaScript API. For example, host a simple HTML page in a PaletteSet for user to enter login info and submit the licence valication info. Of course you can do it entirely silently in the background, as long as your plugin knows where the get the login credential (ask user input when the plugin is used the first time, maybe).
Azure provide one month free host trial. Or if you are MSDN subscription member, you can have a free access to Azure with certain resources limits. Well, if you have not done a web app (based on ASP.NET, I assume) before, one month free access to get your own web service for licence validation is certainly not long enough, but the real cost for serious web app hosing is very affordable.