Just like Reno in this comment, Zack isn't magical on his own, but he does have access to the same items and Materia that Reno has.

The difference is, Materia is so, soooo much easier for Zack to get his hands on. He'll find it killing monsters, he'll order it online. It's everywhere.

He's a little less skilled with magic at this point in canon than he will be far later on, though, so his spells at present are limited to Fire, Blizzard, and Cure- A fireball spell, an ice spell, and a healing spell respectively.