It may depend on your exact definition of whether they make it. I believe there are chips with Asus branding (just like I'm certain there are Lenovo-branded chips, and Asus is an ODM). I suspect they design some chips.

However, Asus does not have their own fabs. At its core, this is where the shortage comes from. They would need to either find or build the additional fab capacity, which is a multi-year project.