Can I get yeast infection from using the cup?

There is no known increased risk of yeast infections connected to the use of menstrual cups. You’re more likely to get an infection from bacteria on your hands transferred via the menstrual cup than you are to get an infection from the cup itself (presuming the cup is being used correctly).

Candida is caused by an overgrowth of yeast that occurs naturally. A yeast infection occurs when there is too much of it - this can be triggered by many different things. 

Even though the cup doesn't support bacterial growth, you can transfer bacteria to the cup so if you use the cup while you have a yeast infection, the bacteria stay on the cup and the next time you use it, you can infect yourself again.