I would like more information if you can. What browser are you using? Maybe try using another browser for itch games -- Chrome works well for it.
Your achievements are being saved? Like you have progress and check marks? If the achievements are working then your unlocks should persist as well, so that's very odd.
There is no mid-run save, if that's what you mean. You should be able to complete a run and gain unlocks for the next run. But if you quit on level 3, you have to restart when you come back.