There is a colour picker function for vector objects for stroke and fill but it’s clumsy compared to the Ctrl+click operation of the usual colour picker for paint layers.
When a vector object is selected, it’s Fill can be set by using the FG Select a Colour window colour picker and the Stroke by the BG colour selection, by selecting a colour from somewhere else on the canvas.
If you want to select a colour from another vector object you use Shift+click.
Also, the Select a Colour window is quite large and often has to be dragged out of the way to see the location you want to pick a colour from. As I said, it’s clumsy.
There may be other or better ways that I’ve missed.
You ought to create a Feature Request topic for this. My idea would be a colour picker icon in the Tool Options docker in the same way that the FG/BG Select a Colour window has a colour picker icon.