Migrate injected client to package:web

Author: derekxu16

dwds/CHANGELOG.md

@@ -1,5 +1,7 @@
 ## 24.2.0-wip
 
+- Migrate injected client code to `package:web`. - [#2491](https://github.com/dart-lang/webdev/pull/2491)
+
 ## 24.1.0
 
 - Fix bug where debugging clients are not aware of service extensions when connecting to a new web app. - [#2388](https://github.com/dart-lang/webdev/pull/2388)