Can I use TeamCity with your hosting?

Yes!

You must use TeamCity with the Web Deploy configuration.

You can find an example of how to do this on the Jetbrains website: https://www.jetbrains.com/help/teamcity/continuous-delivery-to-windows-azure-web-sites-or-iis.html 

We do not have and cannot install the TeamCity deployment agent on our servers.

To get started please open a support ticket from within your My.UmbHost account.