I have not personally have not had any experience connecting to the Synthesio API. I do have knowledge about the available options on the Tableau side. For direct connections, if the data is available over HTTP, the connection to data source can be made with the WDC(Web Data Connector). I have included several resources for using the Tableau WDC below. The first link is to the Tableau Online Help on the WDC feature. Additionally, I included links to the specific section in the community and the Tableau GitHub section on the WDC.
Tableau can also connect to a myriad of files if the program can create a .csv of the data source. As well as many times there are options to configure programs like Synthesio to do nightly uploads to a database. (I would suggest a Tableau supported one) Then connect to the database from Tableau.
Hope this helps!