「~~させ (causative verb form) + て + いただく」 expresses receiving the permission (or opportunity) to perform an action from another person.
「いただく」 = 「もらう」 in meaning. Former is only politer than the latter.
「[取]{と}らせていただいた」 means "I/We received the permission to take/collect ~~."
One could also use as a translation "I/We had the pleasure of taking/collecting ~~."
*Note that I personally do not write supplementary verbs (in this case, いただく) in kanji. That is actually the rule both in schools and the printed media.
「その[時々]{ときどき}で取らせていただいた[先生方]{せんせいがた}のデータです。」, therefore, means:
"These are the data from (or "on") the teachers that I/we had the pleasure of collecting at different times."
(Without further context, one could not tell if it is "from" or "on/regarding".)