Yes, you can do this in theory. The very simplest form of the idea I know of isn’t even technically that high tech, just very large in terms of manufacturing scale and also very slow (even by slower than light star travel standards).

You’ve probably heard of the concept of a solar sail, where a spacecraft with a very big, thin reflective foil sail is pushed by the sun. If you design the size and mass of such a spacecraft right, you can make it’s outward acceleration when pointed directly away from the sun equal the inward acceleration due to gravity. Add some means to control the orientation of the sail to keep it pointing directly outwards, and it can be made to stay balanced, staying in one spot, not actually in orbit, just hovering.

Say we make a lot of these. We figure out how close to the sun we can hang them without them overheating, and we cover one side of the sun, almost completely, in these sail craft (the actual amount of material, though large, might not be that insane even, if the sails can be thin enough. Not compared to the mass of a star anyway). They don’t have to be attached together into a rigid shell, just a big swarm of them whose net effect is to stop most of the light on one half of the sun and reflect it to the other side. We can leave gaps between along the orbital paths of any planets we want to still have light.

The swarm of sails will stay bound to the sun, and the combined sun-sails system now emits light but biased in one direction on average. This makes it basically a form of rocket. It’ll be extremely slow to accelerate on account of the extreme mass of the sun compared to the light and such coming off, but in space even a small acceleration adds up over time.

This is called a shkadov thruster. There are other, more complicated but probably faster to accelerate ways you could hypothetically use the sun to move itself, but I’m less familiar with how they work. The wiki page on stellar engines might give you ideas.

This is probably not a great way to go colonize other planets though. The sheer scale of manufacturing capability needed to do this implies enough resources and energy to launch much faster spaceships, and even a spaceship with spinning habitation rings so big or numerous as to fit an entire self sufficient country inside would be much easier to move around than the sun. You could even use the sun to move the ship instead of moving the sun (for example, take that swarm of solar sails, cover the whole sun instead of just one half, and include solar panels to turn a lot of the sunlight into usable energy. Then run big lasers off the sunlight, and point them at another sail attached to the ship, so that it’s like a solar sail but with way more light pushing on it than it would normally have). You could send out probes to do this to other stars and bring them closer to other stars though, or even hypothetically send out a very long term mission where a probe that can build copies of itself goes to another galaxy when it gets there, builds copies and sends them to nearby stars, which copy themselves again, until every star in that galaxy has the probes, and then the probes build the mirror swarms and direct all the stars in the new galaxy to fly towards your original one. It’ll take them a very long time, millions of years at the very fastest and closest, to get to you, but “running out of materials to the point that you need to harvest entire other galaxies to satisfy your needs” is such a far off problem that that’s probably not an issue.