I managed to do this by working for a company that manages and provides crew for a range of vessels, I went from bulkies to RO-PAX then to tankers. I had to take a demotion sail as a 3/O whilst understudying for my tanker endorsement, which meant the chief officer had to watch over me during cargo ops for a few months.
If you find a company like this you’ll need to serve some time on dry ships first whilst proving to the company that you’re a good, professional operator that they should consider for a change to tankers, conduct reports and the like from captains and chief mates that you sail with will help greatly.
Well worth doing if you can find a way, the cargo work can be challenging (which I enjoy) and I find them to be more comfortable at sea when loaded than most dry ships. The only downside I find is the amount of over-regulation and paperwork that constantly builds and is starting to become detrimental to the operation of these vessels.