He's using Stormlight, as in Cosmere standards, it is just a different form of Breath with different rules. The common term applied to the energy he's actually using is investiture.
Investiture is part of all the magic systems on each planet shown in the Cosmere. It's part of the metal in the Mistborn series, the Dor in Elantris, as well as the Breath and Stormlight. It's just that in each of those places, it functions very differently. Vasher wouldn't be able to just eat metal on Scadrial to get what he needs, he'd have to burn it like a Mistborn would, which releases the investiture and channels it into Allomancy. Likewise, I doubt he could breath in runes near Elantris, although it's possible that one of the Elantrians spells could do what is necessary to just release the raw power into him.
Stormlight is easy enough to access on Roshar - probably easier then Breaths are in Hallandren. So he's subsisting off that.