Sanjay Bala wrote:

I have a heating duct that empties up into the room at the perimeter of the
house where it is cantilevered 18" over the 1st floor.

The duct ending is run between the two joists and is missing insulation.
Can anyone think of a way to cut from inside the register opening to put
insulation into the cavity? There isn't much room for a drill, even a
sideways drill.

The only other option is to drill/cut from underneath outside. But that
would involve removing the vinyl soffits, etc.

Any ideas would be appreciated.

Is the ceiling finished in the 1st floor room under the cantilever
section? 18" isn't very far to reach and stuff insulation if you can
access the joist cavity from the 1st floor room.

If the register is fed from flexible duct you might be able to move it
back and access the cavity from above, but flexible duct is also usually
insulated already.

Otherwise, removing the soffit under that area is going to be your best
bet.