I am changing the location of my kitchen sink to an island that will be 9 feet from the old drain and vent, will the sink still drain properly and vent at this distance as the island is in the middle of the room and I can not run a new vent.
Thanks.

You do not wet vent a island installation. You loop vent it,(see image)
As the image shows, you run a drain line from the old kitchen vent, loop it up making your sink stubout 18" off the floor on the drain line,( the bottom pipe).
The top line loops back and ties into the old kitchen vent above the drain line connection. Another option, if local codes allow, would be to run the drain line up to a tee, stub out from the tee and install a AAV ( Air Admittance Valve or cheater vent) on the tee. Good luck, Tom