In this answer, Darius Jahandarie says that ないで is a te-form. I asked about this on chat, and Flaw answered with a question:
If auxiliary ください follows after てform
and ないでください exists, can we reversely conclude ないで is て form?
This makes sense to me, so I think it is a te-form. But a te-form of what? It doesn't look like the te-form of ない, which I think is /nak-u-te/.
Etymology would be helpful here, but I found a comment by Matt saying the etymology of ないで is basically unknown, and another comment by Tsuyoshi Ito pointing to the entry in 大辞林, which seems to agree that there is no established theory for the origin of ないで.
So perhaps the best thing to do is to focus on how the word functions and ignore etymology. Two possibilities come to mind:
Perhaps ないで is a word with only one form, and that form happens to be a te-form.
Perhaps ないで is considered an alternate te-form for ない, even if that's not what it was etymologically.
Do either of these explanations make sense? Is there a better way to explain ないで?