I would like to know how to run an SSIS package from Azure Synapse Studio or whether it's at all possible. Apparently there is no support for SSIS runtime in Synapse but I suppose there should be a way to run legacy packages since there was in Azure Data Factory (SSIS runtime). The only documentation I found about it is this one highlighting the differences between ADF and ASA: https://learn.microsoft.com/en-us/azure/synapse-analytics/data-integration/concepts-data-factory-differences but couldn't find any documentation explaining the reasons, or how to run legacy packages with Synapse.
Please, try to downaload the Azure Feature Pack for integration services.
You can find some useful hits and task to work in azure synapse with dtsx packages.
SSIS IR support in Azure Synapse Analytics is now in preview:
You should still consider refactoring your packages to Synapse Pipelines as although you can now run SSIS in Synapse, you do inherit some of the issues with SSIS. For example SSIS has never been great at things like dynamic columns. If you were importing 10, or 30 or 1,000 tables you would need 10, or 30 or 1,000 Data Flow tasks. In ADF / Synapse pipelines you would just a For Each loop (which runs in parallel), a single Copy activity and appropriate metadata. ADF / Synapse Pipelines also have better support for modern data sources parquet, delta etc
As per the documentation, Azure Synapse Analytics does not support the Azure-SSIS Integration runtime: To answer your question directly there is no way to run a legacy package from Synapse Studio. Therefore your choices are to either use Azure Data Factory (ADF) which does support the SSIS-IR (but brings some duplication if you're already using Synapse) OR refactor your SSIS packages to native Synapse pipelines.
as per documentation, SSIS packages now available to be run in Synapse via the SSIS Runtime available in the Management section of Synapse Studio. https://learn.microsoft.com/en-us/azure/data-factory/create-azure-ssis-integration-runtime