I wrote: Please let me know when would be a good time to...
Correction: Please let me know when a good time would be to...
OK, I got it wrong above so I should post a more detailed (and better) answer. I wasn't totally wrong -- subject-auxiliary inversion is still involved, but so are other rules, which I didn't notice.
This OQ uses Please let me know as the matrix clause with the embedded question complement. I've tried it with several others and as long as it's an embedded question, the phenomenon's the same, so I'll continue to use it. On the other hand, "to ..." is an uninspiring infinitive, so let's use to leave, which fits many contexts. And let's put a subject back on the infinitive, just to keep things straight; I'll use us as the subject (again, fitting many contexts), and add the for subject marker to go with the to verb marker of the infinitive complementizer. Viz, with bracketed clauses:
- [Please let [me know [when would be a good time [for us to leave]]]].
Perfectly grammatical, but irregular. There are two auxiliary verbs (would and be) in the question, and both of them are inverted, instead of just the first one:
- Please let me know when would a good time be for us to leave
Both of these are not normal embedded questions, which don't invert auxiliaries. Without inversion, we get the following
- Please let me know when a good time for us to leave would be.
as well as the following, with Extraposition from NP detaching the infinitive
- Please let me know when a good time would be for us to leave.
from the subject NP a good time for us to leave.
Other possibilities abound; What time (or often just What, with time implicit in a good time), for instance, produces a different cascade of possibilities.
But not all of them are grammatical, or at least regular -- though modals are such a charley foxtrot it's hard to distinguish. For instance, there's only one variant (with would fronted but not be) that allows extraposition it in the question:
- When would (it) be a good time for us to leave?
- When would be (*it) a good time for us to leave?
- When would (*it) a good time for us to leave be?
Though varieties of embedded clauses allow Extraposition:
- Please let me know when (it) would be a good time for us to leave
This is a fairly sticky, though easily constructed, problem, and it's not surprising that an online free service didn't get it all.