As dderolph pointed out, it depends on the hoster, but you would likely have a problem getting Teamspeak on any hoster's platform without their cooperation. Teamspeak is a binary executable service that has to be installed as an application on the server where you want to access it. What you typically have the freedom to do is upload HTML, style sheets, images, libraries, and other types of files as content to run under the web server application. Some of this can be executable, such as .Net binaries that run under IIS, but it all executes within the context of the web server. You do not typically have the freedom to install additional server applications. To get that you move to a higher tier managed hosting environment that is much more costly.
But if all you want is Teamspeak you can get a TS server for $10-15 a month, for up to 25 users or more.