If you have the opportunity, visiting this museum is highly recommended - they let you walk inside the ship and see both the historical context, as well as the modern restoration process.
It's located in a beautiful park and the nearby Nordiska museum is also quite impressive.
I've been visiting the Vasa every few years for 3 decades and there has never been a time when museum goers were allowed onto or into the ship. I haven't been to Stockholm since the start of the Covid pandemic but I very much doubt that has changed.
They do have some rooms on the side with mock-ups of some of the ship's quarters - maybe that is what you are remembering?
It's funny because I also had this memory of walking around inside the ship when I visited the museum as a child. A few years ago when I visited again I realised it must had been the interior mockups I remember.
[1] https://www.vasamuseet.se/en/