Is it possible to run out of cards in Scrolls? I don't see a deck count or anything during battle so I wasn't sure if it was something that I should keep in mind.
No, you can't.
I was under the assumption that yes, you should be able to deck out but that assumption was corrected on the last game I played. While trying to figure out the best strategy for Order when playing against an opponent with a board filled with lobbers, I found that I could actually "bank" cards in my hand because lobbers don't hurt your totems.
I never found a good strategy but I did notice that my single, Honorable General appeared twice in my hand. First, near the beginning of the game before I started banking and much later, after I had cycled through more than the 50 cards in my deck.
No, you can't run out of scrolls.
At the start of a game all your scrolls are in your deck (a.k.a. the library). After scrolls are drawn or sacrificed they end up in the graveyard. When there are no more scrolls to draw from the library, the graveyard is shuffled and reused as a new library.
This is described in the wiki.