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We present a search for gravitational waves from inspiraling, planetary-mass ultracompact binaries using data from the first part of the fourth observing run of LIGO, Virgo, and KAGRA. Finding no evidence of such systems, we determine the maximum distance reach for such objects and their merger rate densities. Then, we identify classes of primordial black hole mass distributions for which these rate limits can be translated into relevant constraints on the mass distribution of primordial black holes, assuming that they are all composed of dark matter, in the mass range <inline-formula><mml:math><mml:mrow><mml:mo>[</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>]</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>⊙</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>. Our constraints for the class of primordial black hole mass distribution functions <inline-formula><mml:math><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula> we consider reach down to <inline-formula><mml:math><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo><mml:mo>∼</mml:mo><mml:mn>0.1</mml:mn></mml:math></inline-formula>. They are consistent with existing microlensing results in the planetary-mass range, provide a complementary probe to subsolar mass objects, and are publicly available [1].