Haben Sie irgendwelche Fragen?

English Translation

Do you have any questions?

This is the first time I’ve run into a specific German word for the “any” in English sentences of the sort, "Are there any questions?, “Do we have any eggs?” Is there a time and/or place I’m not aware of, where something like “irgendwelch” is required?

I think that “irgendwelche” is actually a bit more than just “any”. It’s more like “any kind/sort of”. So with questions this works, but with eggs it would sound a bit weird. I would just say “Haben Sie Eier?” or maybe “Haben Sie Eier da?”, where the “da” means “available” or “in stock”.

Also note that “irgendwelch” is not really a word (*). It’s at least “irgendwelche”, because it’s usually plural where it’s gender-independent, or sometimes “irgendwelchen” if it’s masculine singular, or “irgendwelches” for neuter, and “-er”, “-em” for various cases.
For example: “Ich habe keine Lust auf irgendwelchen Stress.”

Funnily, the english wiktionary page doesn’t have an inflection table, but the german one does:
irgendwelche – Wiktionary

(*): I know duden.de has “irgendwelch”, because that is technically the base form, but you will never come across it, because it will always appear in some kind of inflected form. Neither the english nor the german wiktionary have an entry for “irgendwelch”, only for “irgendwelche”.