Our organisation is interested in creating our new website using Umbraco, but we are unsure if our CiviCRM system would be compatible. Does anyone use CiviCRM with Umbraco or know if they work together?
Umbraco won't work directly with civicrm (they are written in different programming languages, ASP vs. PHP)
What you can do easily is to host your civicrm with one of the supported CMS (drupal, wordpress or joomla) into a subdomain (eg. members.example.org) and switch the main site to umbraco (www.example.org)
the civi site will not use any CMS feature beside handing the login/password, but might want to customise a bit the layout so it looks more or less like your main website (eg set the colors, put your logo), so you can use some civi pages (like donation, membership registration, event registration...) into that sub site without confusing too much your visitors, and add a link to the civi subsite when needed in your main site.
It's not as transparent and nice as having everything on the same site, but plenty of organisations do have different sites for the CRM and the main site, and it seems to work for them
There is no out of the box integration between CiviCRM and Umbraco, only for drupal, joomla and Wordpress.
You can use CiviCRM Profiles to do some (simple) integration between CiviCRM and any Website, e.g. for Newsletter-Signup. You can also create a custom function or forms in Umraco that will exchange data with CiviCRM using CiviCRM's REST API but that requires coding/customisation.