I am just beginning the vegan lifestyle. One of my main motivations is whether this diet could reduce my migraines. Does anyone have any advice or experience with migraines and the vegan diet?

Not just vegan, it should be clean whole plant diet to provide the most health benefit.
Substitute processed carbs with whole grains and beans.
Minimize the consumption of vegetable oils as much as possible, instead eat fats from whole foods like seeds, nuts and avocados. Lots of greens and vegetables. This will reduce the migraines.