There is an abundance of good information on bunions online, so there is no point going over all that information again.
The running shoes that are best if you have bunions are those that are wider in the forefoot to accommodate the bunion. The most typical running shoes that are made on a wider last is often not wide enough. You could try on the wider last models of each of the brands to see how they feel on you in the store. There is really only one brand that is any good here and they are the Altra running shoes. They are very wide in the forefoot compared to other brands and they have the width to accommodate the enlarged joint that is associated with the bunion. If you have a bunion, then try a pair in a running shoe store to see if they are for you.
What else could you do if you have bunions?
The only way to get rid of bunion is with surgery, however, there are other options to help with the symptoms. What is important is to keep the joint supple and flexible with exercises as it does have a tendency to become stiffer or more rigid. A bunion corrector can also be used to help with that somewhat. If you can not get the shoes right, then there are pads chapped like donuts that can be worn to keep the pressure off the joint.