No, you cannot run TurboTax software on a Chromebook. You have to use a Windows computer or a Mac. (And you have to purchase the correct download for the computer you are using - either the Windows download or the Mac download.)
Here are the system requirements for the Windows and Mac software.
https://ttlc.intuit.com/community/system-requirements/help/minimum-system-requirements-for-turbotax-windows-software/01/26175#M512
https://ttlc.intuit.com/community/system-requirements/help/minimum-system-requirements-for-turbotax-mac-software/01/25609#M686