We search for gravitational-wave transients associated with gamma-ray bursts (GRBs) detected by the Fermi and Swift satellites during the first part of the third observing run of Advanced LIGO and Advanced Virgo (2019 April 1 15:00 UTC–2019 October 1 15:00 UTC). A total of 105 GRBs were analyzed using a search for generic gravitational-wave transients; 32 GRBs were analyzed with a search that specifically targets neutron star binary mergers as short GRB progenitors. We find no significant evidence for gravitational-wave signals associated with the GRBs that we followed up, nor for a population of unidentified subthreshold signals. We consider several source types and signal morphologies, and report for these lower bounds on the distance to each GRB.