We present a search for subsolar mass ultracompact objects in data obtained during Advanced LIGO's second observing run. In contrast to a previous search of Advanced LIGO data from the first observing run, this search includes the effects of component spin on the gravitational waveform. We identify no viable gravitational-wave candidates consistent with subsolar mass ultracompact binaries with at least one component between 0.2 Ma-1.0 Ma. We use the null result to constrain the binary merger rate of (0.2 M, 0.2 M) binaries to be less than 3.7×105 Gpc-3 yr-1 and the binary merger rate of (1.0 M, 1.0 M) binaries to be less than 5.2×103 Gpc-3 yr-1. Subsolar mass ultracompact objects are not expected to form via known stellar evolution channels, though it has been suggested that primordial density fluctuations or particle dark matter with cooling mechanisms and/or nuclear interactions could form black holes with subsolar masses. Assuming a particular primordial black hole (PBH) formation model, we constrain a population of merging 0.2 M black holes to account for less than 16% of the dark matter density and a population of merging 1.0 M black holes to account for less than 2% of the dark matter density. We discuss how constraints on the merger rate and dark matter fraction may be extended to arbitrary black hole population models that predict subsolar mass binaries.