Ultralight axionlike particles (ALPs) can be a viable solution to the dark matter problem. The scalar field associated with ALPs, coupled to the electromagnetic field acts as an active birefringent medium, altering the polarization properties of light through which it propagates. In particular, oscillations of the axionic field induce monochromatic variations of the plane of linearly polarized radiation of astrophysical signals. The radio emission of millisecond pulsars provides an excellent tool to search for such manifestations, given their high fractional linear polarization and negligible fluctuations of their polarization properties. We have searched for evidence of ALPs in polarimetry measurements of pulsars collected and preprocessed for the European Pulsar Timing Array (EPTA) campaign. Focusing on the twelve brightest sources in linear polarization, we searched for an astrophysical signal from axions using both frequentist and Bayesian statistical frameworks. For the frequentist analysis, which uses Lomb-Scargle periodograms at its core, no statistically significant signal has been found. The model used for the Bayesian analysis has been adjusted to accommodate multiple deterministic systematics that may be present in the data. A statistically significant signal has been found in the dataset of multiple pulsars with common frequencies between 10-8 and 2×10-8 Hz, which can most likely be explained by the residual Faraday rotation in the terrestrial ionosphere. Strong bounds on the coupling constant gaγ, in the same ballpark as other searches, have been obtained in the mass range between 6×10-24 and 5×10-21 eV. We conclude by discussing the problems that can limit the sensitivity of our search for ultralight axions in the polarimetry data of pulsars, and possible ways to resolve them.