What are the odds of a 97 year old having breast cancer?

It's possible: There is no upper age limit for any cancer.

Not high: The median age of breast cancer in female in us is about 67 years. That means the further you are away from the median the lesser is the chance. But it's still a possibility. I would not recommend routine mammogram in this age group. Just use common sense and if one finds a lump that's new, then medical attention should be sought.