The Short-Baseline Near Detector time projection chamber is unique in the design of its charge readout planes. These anode plane assemblies (APAs) have been fabricated and assembled to meet strict accuracy and precision requirements: wire spacing of 3 mm +/- 0.5 mm and wire tension of 7 N +/- 1 N across 3,964 wires per APA, and flatness within 0.5 mm over the 4 m +/- 2.5 m extent of each APA. This paper describes the design, manufacture and assembly of these key detector components, with a focus on the quality assurance at each stage.