> I was wondering if you could solve this with a camera flash

Yeah, I think you can - there's been a few "multi-flash 3d reconstruction" approaches proposed in the last 5-10 years. e.g. [1], possibly also [2]

[1] https://www.nature.com/articles/srep10909 [2] https://www.researchgate.net/publication/220829727_Shape_fro...