What is Scalise sniffing throughout the movie?

Yep. It's a small nose inhaler, used for stuffy noses. I think the point here is not that he has a cold, but (as you can learn from reading Jack Kerouac) back then those things contained Benzedrine, a mild but effective stimulant (speed). The Beats used to open them and dip the Benzedrine-soaked sponge inside in water, to create a speed-tea. But I imagine sniffing it all day would do the trick also.

Eddie Muller on TCM seems to indicate that the character was asthmatic.