For each player, any face-down writing surface will do.
The standalone version of Dixit Odyssey's 12 player boards are annoying to use. Unfortunately the face down chits that come with the basic Dixit version are numbered 1 through 6 to accommodate 6 players only. For games of 7 or more players, consider replacing these chits with an opaque writing surface. Players can write their numbered guess on it, then turn it face down and rest their writing utensil on it. The storyteller can see this, and call out, "Pencils down, everyone?" as necessary to move things along.
Dry erase boards and markers work spectacularly for this purpose. I have used the colored dry erase boards from Wits and Wagers, as well as the dry erase pads from Telestrations Party pack. Players don't even need to have their own colors and those colors don't need to match their bunny color (and whatever extra pieces are acting as player tokens for the additional players). When scoring players can move thier own piece or tell someone to move their piece - confusion over that is minimal.
For people who own the standalone Odyssey version of Dixit, that has the board with 12 numbered slots for cards so this next part isn't necessary. For people who have the basic Dixit (with expansions hopefully) they can again use use any writing surface to represent slots 7-12 when displaying the additional cards beyond 6.