There is no particular supplication established in the Sunnah when drinking Zamzam. However, Abdullah bin Abbas (may Allah be pleased with him) used to recite the following after drinking Zamzam water. [1]
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* This ‘supplication’ post/sheet only covers what should be said and does not address the actions to be taken (if applicable).
Ref
“Getting the Best out of Al-Hajj (Pilgrimage)”, By Abu Muneer Ismail Davids, Published by Darussalam, Second Edition – August, 2006, Chapter 6, pg 221 [10004]