I have updated my answer based on several comments I've received.
Although it seems that the word suggester does actually exist as a variant of suggest (per Merriam-Webster), I find it so uncommon that, while understandable, it would give most people some pause on reading it.
I would say that a more common word is proposer.
[Merriam-Webster, propose]
1 : to form or put forward a plan or intention
// man proposes, but God disposes
Other Words from propose
proposer noun
In the right context, such as fields to be filled in on a form, the word would be quite appropriate:
Proposal: (____)
Proposer: (____)
Approved: (Yes / No)
Also, I've been informed that proposer is frequently used in formal debates.
As for comparing suggester to proposer, Google Ngram Viewer indicates that proposer is far more common, at least in print:
![suggester versus proposer](https://i.stack.imgur.com/Iy5G5.png)
As I final note, I will add that while proposer might sound strange to some, so too would asker and poster, as used in the question. Just as it would be more common to say the person who asked or the person who posted, it would also be more common to say the person who proposed (or the person who suggested). None of these single nouns are as common as a longer phrase; however, proposer is still relatively much more common than suggester.
Best Answer
Whose is absolutely normal there. in writing, and I would use it in speech as well.
Colloquially, you could say "a file [that] I don't know the name of".