C. knew her family well enough to have owned a large photograph of them (which would not have been cheap in the Victorian Era) and kept it around for more than ten years, and
D. Had a reason to keep thier identity hidden
The only person in-universe I can think of who even comes close to fitting those characteristics would be Lewis Caroll/Charles Dodgson, but that still leaves the question of why he chose to send the gift anonymously.
It could have been Radcliffe, as he also kept her rabbit all that time. Or maybe the doctor from the asylum? He did care about Alice and wanted her to get better. It would have to be someone that wasn't that close to Alice, because why else would they not sign their name, but cared enough to send her the picture.