That was generally the most common method during the UnCivil War and to a lesser extent during the Mexican War - where most cartridges initially came prepared for the soldiers to use. However in the Mexican War and earlier periods, it was common for the Artillery to be placed in charge of powder, lead and molds, balls, special cartridge paper, cartridge forms, powder measures, etc. AND make cartridges "in the field" under the direction of Artillery Officers and/or at least Artillery NCO's - when more cartridges were needed. This because the Artillery was properly seen as having the most experience with safely transporting and keeping powder. Oh, they also normally kept the prepared cartridges (when those were available) and issued them to the Infantry as ordered/needed.
Gus