During the first half of the fourth observing run (O4a) of the International Gravitational Wave Network (IGWN), the Zwicky Transient Facility (ZTF) conducted a systematic search for kilonova (KN) counterparts to binary neutron star (BNS) and neutron star-black hole (NSBH) merger candidates. Here, we present a comprehensive study of the five high-significance (FAR < 1 per year) BNS and NSBH candidates in O4a. Our follow-up campaigns relied on both target-of-opportunity observations (ToO) and re-weighting of the nominal survey schedule to maximize coverage. We describe the toolkit we have been developing, Fritz, an instance of SkyPortal, instrumental in coordinating and managing our telescope scheduling, candidate vetting, and follow-up observations through a user-friendly interface. ZTF covered a total of 2841 deg\(^2\) within the skymaps of the high-significance GW events, reaching a median depth of g~20.2 mag. We circulated 15 candidates, but found no viable KN counterpart to any of the GW events. Based on the ZTF non-detections of the high-significance events in O4a, we used a Bayesian approach, nimbus, to quantify the posterior probability of KN model parameters that are consistent with our non-detections. Our analysis favors KNe with initial absolute magnitude fainter than -16 mag. The joint posterior probability of a GW170817-like KN associated with all our O4a follow-ups was 64%. Additionally, we use a survey simulation software, simsurvey, to determine that our combined filtered efficiency to detect a GW170817-like KN is 36%, when considering the 5 confirmed astrophysical events in O3 (1 BNS and 4 NSBH), along with our O4a follow-ups. Following Kasliwal et al. (2020), we derived joint constraints on the underlying KN luminosity function based on our O3 and O4a follow-ups, determining that no more than 76% of KNe fading at 1 mag/day can peak at a magnitude brighter than -17.5 mag.