To request "signed photographs" vs "signing photographs"
Though fans continued to hound her with requests for ____ (SIGN) photographs
The Google Ngram that you should use is the one with the preposition "for": for signed photographs,for signing photographs See https://books.google.com/ngrams/graph?content=for+signed+photographs%2C+for+signing+photographs&year_start=1800&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cfor%20signed%20photographs%3B%2Cc0
You will note that there are no results for "for signing photographs".
If we consider "Though fans continued to hound her with requests for SIGNING photographs", it is difficult to see what function "SIGNING" performs. It is not a participle or a gerund acting as a adjective as that would mean "photographs that are associated in some way with the action of signing".
You may object and say "Well, we can say 'This is the pen for signing the photograph'.", but here "for" introduces an instrumental purpose. (Compare, "This is a walking stick.")
But "fans continued to hound her with requests for _____ (SIGN) photographs" is a request for an object, i.e. photographs (of her) that were signed (by her); it is not the purpose of the object.
You should realise that preposition + substantive = modifier, and that "for ______ (SIGN) photographs" must be modifier, and thus that " ______ (SIGN) photographs" must be a substantive.
The constructions are
This is a request to do an action, which requires an infinitive: "I requested her to sign the photograph."
This is a request for something, which requires a substantive. "This is a request for a signed photograph." You will know that a substantive can be a noun phrase and that in your example "signed photographs" is a noun phrase.
We need to ask what it was that the fans did.
They hounded her.
How did they hound her?
They hounded her with requests - "with requests" = preposition + substantive = modifier (here it is adverbial)
What sort of requests?
They hounded her with requests for photographs.
What sort of photographs?
They hounded her with requests for signed photographs. - Signed is an adjective. "signed photographs" is a noun phrase" Noun phrases are substantives.
"for signed photographs" = preposition + substantive = modifier (here it is adjectival). And, all together, {with [requests for signed photos]. } is adverbial.
They did not request her "to sign photos", they requested an item - a signed photograph, i.e. a photograph (of her) that had been signed (by her).
They did not request that she do something, i.e. sign some photographs, and so "signing" (a participle or gerund that indicates an action) must be wrong.
I think you are correct, but there a slight difference in meaning.
Requests for signed photographs focuses more on a photograph with a signature on it, possibly already done.
Requests for signing photographs (signing photographs being a gerund phrase) would imply there is a number of photographs that she would sign in front of her fans.
Because in this case fans are writing to her (it mentions letters went unanswered), and this is not a press event where she meets fans face-to-face and she could sign the photographs in front of them, I think signed is more appropriate.
I hope that helps clarify the difference!