ACLU: <https://www.aclu.org/>

EPIC (Electronic Privacy Information Center): https://www.epic.org/

Further lists:

Data privacy advocates and associations <https://privacybee.com/data-privacy-advocates-and-associatio...>

Privacy Focused Organizations You Should Know About <https://identityreview.com/18-privacy-organizations-you-shou...>

Privacy & Information Law Research Guide <https://guides.ll.georgetown.edu/c.php?g=468955&p=3962183> (oriented more at research but some overlap)