Our last planet, Neptune, fits neatly within one light-month. The rest of the heliosphere does too, but not the Oort cloud. Looks like empires are purely solar system affairs. Sci-fi authors take note.
Supplying a ground based force follows more of a square-cube law. You need to feed your horses to transport your food to feed your soldiers. Going further means more food for the horses to carry the food for the horses to carry the food for the soldiers. There is an excellent post here that explains this far better than I ever could: https://maximumeffort.substack.com/p/the-tyranny-of-the-wago...
Any limitations on space based empires, while similar, are going to follow somewhat different rules depending on the tech involved. Unlike horses, once you launch you don't really have an opportunity to resupply a space based force going a long distance without inordinate energy expenditure. I'm not convinced force projection over a galaxy would be a problem for a civilization capable of travel at a significant fraction of the speed of light. Solving one problem (travel) necessarily solves the other (distribution of force).
I think the bigger question is more, why bother? What possible reason could a civilization have to go to war at that scale? Human civilization would need to change so much to get to that point that I'm not certain these are questions are explorable at our place in the timeline. We'd be like Romans dreaming about a better abacus.