Do you guys think it's possible to make an "All Sigil, no item run"?

Most likely not, because some of the stars are in hard-to-reach places.

I have actually gone though the game and attempted to beat it without unlocking any items. (This was before the tower skip was found. With the exception of the third puzzle from A4, I was able to collect all tetromino pieces.

The stars are a different story, however. With our current understanding of the game, there is simply no way to get to stars such as B4, B5, B7, and the lower Nexus.