Context. We present the second public data release of the VISTA EXtension to Auxiliary Surveys (VEXAS), where we classify objects into stars, galaxies, and quasars based on an ensemble of machine learning algorithms.Aims. The aim of VEXAS is to build the widest multi-wavelength catalogue, providing reference magnitudes, colours, and morphological information for a large number of scientific uses.Methods. We applied an ensemble of thirty-two different machine learning models, based on three different algorithms and on different magnitude sets, training samples, and classification problems (two or three classes) on the three VEXAS Data Release 1 (DR1) optical and infrared (IR) tables. The tables were created in DR1 cross-matching VISTA near-infrared data with Wide-field Infrared Survey Explorer far-infrared data and with optical magnitudes from the Dark Energy Survey (VEXAS-DESW), the Sky Mapper Survey (VEXAS-SMW), and the Panoramic Survey Telescope and Rapid Response System Survey (VEXAS-PSW). We assembled a large table of spectroscopically confirmed objects (VEXAS-SPEC-GOOD, 415 628 unique objects), based on the combination of six different spectroscopic surveys that we used for training. We developed feature imputation to also classify objects for which magnitudes in one or more bands are missing.Results. We classify in total approximate to 90x10(6) objects in the Southern Hemisphere. Among these, approximate to 62.9x10(6) (approximate to 52.6x10(6)) are classified as 'high confidence' ('secure') stars, approximate to 920 000 (approximate to 750 000) as 'high confidence' ('secure') quasars, and approximate to 34.8 (approximate to 34.1) million as 'high confidence' ('secure') galaxies, with p(class)>= 0.7 (p(class)>= 0.9). The DR2 tables update the DR1 with the addition of imputed magnitudes and membership probabilities to each of the three classes.Conclusions. The density of high-confidence extragalactic objects varies strongly with the survey depth: at p(class)>0.7, there are 11 deg(-2) quasars in the VEXAS-DESW footprint and 103 deg(-2) in the VEXAS-PSW footprint, while only 10.7 deg(-2) in the VEXAS-SM footprint. Improved depth in the mid-infrared and coverage in the optical and near-infrared are needed for the SM footprint that is not already covered by DESW and PSW.