Microsoft have updated the way that Teams is distributed and installed, it is now an app package and there is a new way to obtain Microsoft Teams.
Please make sure that all Windows Store applications are fully updated before downloading the new Teams installer from here.
(This is beacuse there is an internal Windows Application that needs to be fully updated to accept .msix packages)
To do this, go to the Start menu and type in Microsoft Store and press enter, when the Microsoft Store opens click on the ... in the top right corner of the window and click on "Downloads and updates" and then click on "Get updates" which will start the application update process, the Microsoft Store may close itself during this process. You will know that this process has been successful when you relaunch the Microsoft Store and the layout has changed, with a side bar on the left side of the window, with the "LIbrary" button in the bottom left of that window, if this looks like your version of the Microsoft Store, please proceed to follow the guide below.
Obtaint the .msix Teams package from the link above.
Navigate to the downloaded package and double click on it to run it.
Click on the install application button.
That's it, Teams should open when the installation has finished.