Organic electronic ion pumps (OEIPs) are versatile tools for electrophoretic delivery of substances with high spatiotemporal resolution. To date, OEIPs and similar iontronic components have been fabricated using thin-film techniques and often rely on laborious, multistep photolithographic processes. OEIPs have been demonstrated in a variety of in vitro and in vivo settings for controlling biological systems, but the thin-film form factor and limited repertoire of polyelectrolyte materials and device fabrication techniques unnecessarily constrain the possibilities for miniaturization and extremely localized substance delivery, e.g., the greater range of pharmaceutical compounds, on the scale of a single cell. Here, we demonstrate an entirely new OEIP form factor based on capillary fibers that include hyperbranched polyglycerols (dPGs) as the selective electrophoretic membrane. The dPGs enable electrophoretic channels with a high concentration of fixed charges and well-controlled cross-linking and can be realized using a simple "one-pot" fluidic manufacturing protocol. Selective electrophoretic transport of cations and anions of various sizes is demonstrated, including "large" substances that are difficult to transport with other OEIP technologies. We present a method for tailoring and characterizing the electrophoretic channels' fixed charge concentration in the operational state. Subsequently, we compare the experimental performance of these capillary OEIPs to a computational model and explain unexpected features in the ionic current for the transport and delivery of larger, lower-mobility ionic compounds. From this model, we are able to elucidate several operational and design principles relevant to miniaturized electrophoretic drug delivery technologies in general. Overall, the compactness of the capillary OEIP enables electrophoretic delivery devices with probelike geometries, suitable for a variety of ionic compounds, paving the way for less-invasive implantation into biological systems and for healthcare applications.