In Illustrator, go to Preferences > File Handling & Clipboard and in the Clipboard section, enable (check) "On Copy: Include SVG Code". Select your vector(s) in Illustrator then in Hype, paste. As noted, this will create a blank rectangle. Click the rectangle then the edit Inner HTML button (or, from the File menu, select "Edit Element's Inner HTML") and in the resulting popup, paste again. This should paste the SVG vector code from Illustrator without needing to export the Illustrator document as an SVG. A caveat to this is that if your vector in Illustrator isn't at the 0,0 offset, then it will be offset in the Hype rectangle and may not be fully visible.
I had never tried this before and not sure where the change was made. Perhaps Tumult can comment.