Do people with ambiguous genitalia have periods?

Depends. There are many types of genital ambiguity. However, if a person has a uterus and ovarian tissue, then periods can occur even if they seem to have a penis.
Some of them. Provided they have ovaries, uterus and patent vagina. This certainly applies to females with congenital adrenal hyperplasia, and occasional to hermaphrodites.

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