Double click on the smart vector object layer icon (not the layer name.)
This will open in Illustrator.
Vector Smart Object in Photoshop will resize to the resolution of the Photoshop file. So they are vector while in Phothsop(ofocurse they have to be vector originally), but when you print print at the resolution of your Photoshop file.
In Illustrator, File > Open and choose the Photoshop file. In the Open options choose Convert Layers to Objects.
If you want only the paths you can just copy from Photoshop and paste in Illustrator. Or File > Export Paths to Illustrator. There will be no color thought - the vector objects in Photoshop do not have color attributes they are only paths used for masking the raster fill of a layer or for making raster based selections. They've made the Convert Layers to Objects option assigns the color of a shape layer to to the fill of the converted object in Illustrator but this can't be done with copying or exporting paths because you can draw paths without a shape layer in Photoshop.
edit: if I remember, make sure all paths are entirely within the canvas area or they may not show in Illustrator