from the licensing terms at https://docs.microsoft.com/en-us/sql/ssdt/sql-server-data-tools-license-terms

Installation and Use.

  You may install and use any number of copies of the software on your devices to design, develop and test your programs.

And looking at the features per edition at https://docs.microsoft.com/en-us/sql/integration-services/integration-services-features-supported-by-the-editions-of-sql-server

we see that most of the integration services features are not available on the express edition

So based on the above most would consider that you can not use SSDT to do production work at it is not on the list of allowable usage.

As for the link that Phil gave - yes the Integration Services SERVICE is not required to run the packages - this is because from SQL 2012 SSIS catalog was introduced and that does not require the independent service - but the usage of Integration Services is still there and still needs to be selected as part of an install of a instance