Probably not. Even though your period leaves your vagina really moist, and it's warm there to start with, and this is the perfect environment for yeast to grow, since your tampon is absorbing most of the moisture you aren't that likely to get a yeast infection from using a tampon. Yeast infections are pretty common though, so if you're itching a lot, even just on the outside, and you have a whitish, cheesy discharge in your underwear (why is everything gross always described using food?!) you may have a yeast infection. We all have yeast growing in our vaginas and the `good' bacteria that are there keep it in check, but if the balance is out of whack, the yeast go wild! Eat a healthy diet, especially yogurt, and avoid antibiotics unless you really need them and you will decrease your chance of a yeast infection. If you think you might have one, it's a good idea to see a healthcare professional the first time and be sure that's what is going on before just using an over-the-counter treatment. It may be a little awkward to talk about itching and goop in your underwear, but since other things can give the same symptoms it's always good to get checked out.
I've dealt with yeast infections and they're not very fun ... thank goodness they're easily treated! In my experience, tampons haven't been the cause. Moisture, however, is a big culprit behind yeast infections for women. When I said a sad goodbye to my pretty silky panties and switched to a more breathable cotton, I had much less trouble with yeast infections. Antibiotic use is another possible cause of yeast infections. Be sure to read health expert Dr. Molly's answer. She'll give you the lowdown on yeast infections from a medical perspective.
I've never gotten a yeast infection and I'm a regular tampon user. But I'm not completely sure if tampons can or can't give you one, so you should see what health expert panelist Dr. Molly says about it. I do know that moist environments can be breeding grounds for yeast infections, so just make sure you keep everything clean and dry down there. The extra moisture from your period could make it seem like a cozy place for a yeast infection to make a home. I also always shower or bathe every day of my period, regardless.