Is there a way to autoscroll tab or score?

You are great! There is one thing need to know with VST Live.
Any Layer, Stack, Module is inserted to a Song/Part it will be destroyed as soon as exiting that PART.

So if using multiple PARTs / Song would require as much instances of Viewer module… but in this case:
The trick is to create a Global Part and insert the PDF Viewer module in that Global Part. That global part will remain for the global project.

Goto PART / Create Global Part