Disclaimer: so far VizieR does not make any difference between FK5 Equinox J2000 and ICRS (and converting one system to the other has no effect). Positions labeled as J2000 in VizieR can be either FK5 J2000 for old catalogues, or most likely ICRS for recente catalogues. In practice the FK5 system internal error is of the same order as the FK5/ICRS offset (about 80 mas for the latter).
For uploaded files: the user has to choose the positions in the Tables management tab on the cross-match web page.
For VizieR tables which are not in the large tables list (the list appearing when clicking VizieR in the table selection panel): the VizieR computed (RAJ2000, DEJ2000) positions are used. If a table contains proper motions, they are probably taken into account and the position are provided at epoch J2000. In case of doubt, please check the VizieR page of the table with the J2000 and decimal Preferences selected.
For VizieR large tables: the first positional columns of each table in the xmatch ouput are the ones used for the cross-match. For most large astrometrical catalogues, the positions are probably the epoch J2000 positions (possibly computed by the VizieR team). For photometrical catalogues, the positions are the main positions provided in the original table.
Specific case: for Gaia DR2, two tables are now provided
Gaia DR2 (or the vizier identifier I/345/gaia2): the positions computed at epoch J2000.0 (using Gaia DR2 positions at epoch 2015.5 and Gaia DR2 proper motions) are used. This is well suited for cross-matches with galactic objects having positions near the J2000 epoch. Although the positional errors have also been computed at epoch J2000.0, they do not account for the full astrometric covariance martix.
Gaia DR2 J2015.5: the original positions, at epoch J2015.5. Those positions are the most precise (no propagation of astrometrical errors on a 15.5 year timescale) and particulary suit extra-galactic studies or cross-matches with photometric catalogues providing positions at an epoch nearest from J2015.5 than from J2000.0.
Specific case: for Gaia EDR3, two tables are now provided
Gaia EDR3 (or the vizier identifier I/350/gaiaedr3): the positions computed at epoch J2000.0 (using Gaia EDR3 positions at epoch 2016.0 and Gaia DR3 proper motions) are used. This is well suited for cross-matches with galactic objects having positions near the J2000 epoch. The positional errors have also been computed at epoch J2000.0 taking into account the full astrometric information (all 6 parameters and the associated 6x6 covariance matrix)
Gaia EDR3 J2016.0: the original positions, at epoch J2016.0. Those positions are the most precise (no propagation of astrometrical errors on a 16 year timescale) and particulary suit extra-galactic studies or cross-matches with photometric catalogues providing positions at an epoch nearest from J2015.5 than from J2000.0.