Since I found How run flutter 'packages pub run build_runner build' with debug mode in intellij idea? I was wondering how this works in VS Code. Furthermore, I didn't liked the solution with copying the build file. This is how i got it working.
Generating the script:
The first thing to do is executing flutter pub run build_runner build so the files in the .dart_tool folder are generated. My app is called meal_app.

The code generator script is located in .dart_tool/build/entrypoint/build.dart.
Running the script:
The script can be run with dart .dart_tool/build/entrypoint/build.dart build but that is just executing, not debugging the script. For convenient debugging the VS Code launch.json needs to be adjusted.
Create a launch.json
The launch.json file configures the launch configurations in VS Code. To create a launch.json select the debug symbol on the right and create the launch.json file.

Create a launch configuration for build_runner
{
"version": "0.2.0",
"configurations": [
{
// Config 1
},
{
// Config 2
},
{
"name": "Debug Widgetbook Generator",
"cwd": "example/meal_app",
"request": "launch",
"program": ".dart_tool/build/entrypoint/build.dart",
"type": "dart",
"args": ["build"]
}
]
}
cwd: [Probably not required] The app for which build_runner is generating files is called meal_app. The meal_app is located in a subfolder called example. Thats why the cwd property is set in the configuration. If your app is not located in a subfolder, you can omit this option.
args: Set to "build". This is similar to the command flutter pub run build_runner build where build is the argument of build_runner.
program: Since the code generator file is located in the structure mentioned above, the configuration needs to know which file to execute.
name: This is the name of the configuration.
Don't forget to switch to the correct configuration