I have this path. I want the yellow half cylinder to automatically follow my path. I tried to use Shrinkwrap but it doesn't find the path. My end result should be that the cylinder is like the annotations I did.
At its simplest, this could be your curve, set to 'Half' Fill Mode, using its native 'Round' bevel...
.. followed up with Solidify, Bevel, and Subdivision Surface modifiers
The curve can have a fairly nasty habit of flipping its twist.. I've found the most stable method has been to set the curve's 'Twist Method' to 'Minimum', as shown, construct the curve in in its XY, and adjust its tilt, throughout, to -90 degrees.