Do women have y chromosomes?

No, men have an X and a Y, while women have two X's.

{They don't usually, but...

'There is a whole lot more to maleness and femaleness than X or Y chromosomes. About 1 in 20,000 men has no Y chromosome, instead having 2 Xs. This means that in the United States there are about 7,500 men without a Y chromosome. The equivalent situation - females who have XY instead of XX chromosomes - can occur for a variety of reasons and overall is similar in frequency'.}