How it is physically possible to play ONUW with 10 players?
We run into the same issues; generally playing with 7-8 people.

There are a few techniques we use to help:


Players don't keep their own cards immediately in front of them; they keep somewhere in between their own seat and the center of the table. Close enough that it's clear whose is whose.
I try to start the night phase already leaning forward, so that if I don't make noise leaning forward if I need to later.
Even with doing these things, taking actions that involve other players' cards such as seer, troublemaker, or robber, are generally just restricted to choosing a player that is seated close to you. As a seer, I would never attempt to look at the card of a player sitting across the table from me.