Migrate injected client to package:web (#2491)

Author: derekxu16

dwds/CHANGELOG.md

@@ -1,5 +1,7 @@
 ## 24.2.0-wip
-- Replace deprecated JS code `this.__proto__` with `Object.getPrototypeOf(this)` - [#2500](https://github.com/dart-lang/webdev/pull/2500)
+
+- Replace deprecated JS code `this.__proto__` with `Object.getPrototypeOf(this)`. - [#2500](https://github.com/dart-lang/webdev/pull/2500)
+- Migrate injected client code to `package:web`. - [#2491](https://github.com/dart-lang/webdev/pull/2491)
 
 ## 24.1.0