I highly doubt it. Without the PIC you can’t stand your own cargo watch. I had heard that Seabulk was taking people as an extra to get their loads and discharges, but I don’t know if that’s still going on.
With OSG the unlicensed are SIU. You could join them and go out for a trip as OS/AB to get the loads and discharges, but like Oakley says, you would need at a minimum Tankerman Assist for that.