We can say with certainty that you'll max raid battle in the story at least once just to show you how it works.
However, with a tiny bit of speculation, I can theorize that wild dynamaxed pokemon will play a much bigger role in the story.
The cities of Galar are enormous. More specificly, the walls are enormous. Not only do we hardly ever see a walled off city in pokemon, but the walls are much higher than they need to be. We can see from the shots of the wild area that only the tallest buildings are visible over the walls. However there are many towns that don't have walls, like the starting town. So what do the steampunk city and the dragon city have in common?
They're both adjacent to the wild area, where wild dynamaxed pokemon come from. Yes, I posit that sword and shield will go full pacific rim and require you to defend the cities from wild dynamaxed pokemon emerging from portals in the wild area. I'm unsure whether these are Galar natives dynamaxed by the portals, or foreign invaders. They did say that some pokemon can only be found in max raid battles.
