Results are presented of searches for continuous gravitational waves from 20 accreting millisecond x-ray pulsars with accurately measured spin frequencies and orbital parameters, using data from the third observing run of the Advanced LIGO and Advanced Virgo detectors. The search algorithm uses a hidden Markov model, where the transition probabilities allow the frequency to wander according to an unbiased random walk, while the J-statistic maximum-likelihood matched filter tracks the binary orbital phase. Three narrow subbands are searched for each target, centered on harmonics of the measured spin frequency. The search yields 16 candidates, consistent with a false alarm probability of 30% per subband and target searched. These candidates, along with one candidate from an additional target-of-opportunity search done for SAX J1808.4-3658, which was in outburst during one month of the observing run, cannot be confidently associated with a known noise source. Additional follow-up does not provide convincing evidence that any are a true astrophysical signal. When all candidates are assumed nonastrophysical, upper limits are set on the maximum wave strain detectable at 95% confidence, h095%. The strictest constraint is h095%=4.7×10-26 from IGR J17062-6143. Constraints on the detectable wave strain from each target lead to constraints on neutron star ellipticity and r-mode amplitude, the strictest of which are ϵ95%=3.1×10-7 and α95%=1.8×10-5 respectively. This analysis is the most comprehensive and sensitive search of continuous gravitational waves from accreting millisecond x-ray pulsars to date.