Are either top or bottom plates required to be 1 continuous piece of lumber? If it's not required, what is the best way to attach the separate sections; a lapping piece over the top of the joint?
Negative. Bottom plates rarely are in any case, since they're fully supported, though better carpenters tend to splice them on studs.
Simply fasten them to the floor with construction adhesive and/or mechanical fasteners as appropriate. No scab is necessary (though one could help keep things in alignment if not splicing under a stud).