It's not correct. We cannot use "might" and "would have been" together. You can have “... I wonder what their reasons would have been.” or “... I wonder what their reasons might have been.”

1 - Embedded questions use statement word order. Did Joe sell his car? I wonder if Joe sold his car. 2 - Might and would are modals. Use one modal, not two. I wonder what their reasons might have been. Your grammar book certainly covers these topics. If you don't have a grammar book, English Grammar in Use by Murphy is famous. Paper copies are cheap and a google search will find a PDF. See units 29 and 50.

No, the sentence you wrote is not correct. I suggest: I wonder what their reasons might have been.