I believe cooperstown has final say. there were rumors that the Devils Rays were going to 'buy' Wade Boggs to wear their cap, before that players could decide. since then Cooperstown decides, often (coincidentally ) it has come down to players rumored not to want to wear a cap from a Canadian team (chiefly the Expos) as when they no longer existed it was thought that both Dawson and Carter would derive more benefit going forward wearing a cap of a surviving organization.
so I think the decision doesn't rest solely with the player although they would take a player's wishes into account. not sure what they would think about no cap. the plaques also I think indicate a 'primary team' (usually but not always the same as the cap team)
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