Is there a way around to convert from Vertices to Line with GPS attributes to keep them ?

Working with FME 2021.2 mostly here. Have 2022, 2023 and 2024 also but no need to use that.

But your question is a bit hard to understand.

You have a shapefile, but the shapefile does not have geometry (what you call GPS). But it has a column lat and long. So you used a VertexCreator to make point geometry. And now you want to convert Point geometry to Line geometry.

This is indeed done with a LineBuilder.

You can connect the LineBuilder to an Inspector to see if the result is as expected.

If that's true then you can start using a FeatureWriter to write the data into a Shapefile. If you set it to the right parameters it should create a Shapefile containing a Line. But for now we don't even know if the data is what you expect so the error could be anywhere.

It would be helpful to see a copy of your workspace or even a screen capture. You need to create vertices using the coordinates before you use linebuilder and you should assign a coordinate system to the line before you write it to a shapefile.