Will invasive species spread from the wood chips I get?

They could. There's seeds and rhizomes of all kinds floating around all over the place, especially in cities, and they will likely be in your new pile of arborist wood chips. The key is to keep an eye on the pile after you spread out the wood chips, and pull any volunteers that sprout up out of the loose mulch. You can also let your pile 'cook' for a couple of weeks before you spread it out in your yard. This will help kill any viable plant material in the pile.