Gravitational waves from subsolar mass inspiraling compact objects would provide almost smoking-gun evidence for primordial black holes (PBHs). We perform the first search for inspiraling planetary-mass compact objects in equal-mass and highly asymmetric mass-ratio binaries using data from the first half of the LIGO-Virgo-KAGRA third observing run. Though we do not find any significant candidates, we determine the maximum luminosity distance reachable with our search to be of O(0.1-100) kpc, and corresponding model-independent upper limits on the merger rate densities to be O(10^{3}-10^{-7}) kpc^{-3} yr^{-1} for systems with chirp masses of O(10^{-4}-10^{-2})M_{⊙}, respectively. Furthermore, we interpret these rate densities as arising from PBH binaries and constrain the fraction of dark matter that such objects could comprise. For equal-mass PBH binaries, we find that these objects would compose less than 4%-100% of DM for PBH masses of 10^{-2}M_{⊙} to 2×10^{-3}M_{⊙}, respectively. For asymmetric binaries, assuming one black hole mass corresponds to a peak in the mass function at 2.5M_{⊙}, a PBH dark-matter fraction of 10% and a second, much lighter PBH, we constrain the mass function of the second PBH to be less than 1 for masses between 1.5×10^{-5}M_{⊙} and 2×10^{-4}M_{⊙}. Our constraints, recently released, are robust enough to be applied to any PBH or exotic compact object binary formation models, and complement existence microlensing results.