Could a panoramic dental x-ray without a lead apron harm a woman's eggs or any part of her reproductive system?

Non-existent risk: It would be on the same level of exposure as being outside in the sun for the day. There have been no substantiated evidence from any studies that have shown clear cause and effect relationship of dental x-rays and reproductive tissue damage.

Not really but: Its still always best to be draped with a lead apron as a precautionary measure. The newer generation of digital machines use far less radiation than the dinosaurs of yester-year!