Apparently, although both “at the 100th anniversary” and “on the 100th anniversary” are used commonly, using the preposition on is more common according to Google search results. I’d like to ask that whether using at is a grammar error or whether there are special structures in which it is preferred to be used.
I suggest you to use "on" in the sentence "... on the 100th anniversary" as it implies that the author is talking about a particular occasion - a specific day. On is used with days, parts of the days, dates, particular occasions, anniversaries and festivals.
I use the links hereinbelow as references:
on - Merriam-Webster: 3 — used as a function word to indicate a time frame during which something takes place or an instant, action, or occurrence when something begins or is done
on - WordReference: 20 - at the time or occasion of: You have to pay cash on delivery (= at the time delivery is made).
I hope this answers your question well.
At the 100th anniversary impiles that the 100th anniversary is an event, likely one that people can attend. Since it happens at a point in time it can be used that way as well.
I'm trying to get this gift so I can give it to her at her 100th anniversary. (You might be taking her out to dinner, which you are considering an event, and giving her the gift then. Or you might be throwing her a party and will give it to her at the party.)
I'm trying to get this gift so I can give it to her on her 100th anniversary. (You are just saying you're going to give it to her on the day of her 100th anniversary, but no implication about a party or event.)