Zip api docs using the repo under test script. (#116437) (#117093)

This is to move more logic to the repo under test and remove them from
recipes.

Bug: https://github.com/flutter/flutter/issues/115489
diff --git a/dev/bots/docs.sh b/dev/bots/docs.sh
index b8b715c..91ee3d1 100755
--- a/dev/bots/docs.sh
+++ b/dev/bots/docs.sh
@@ -148,3 +148,7 @@
   (cd "$FLUTTER_ROOT/dev/docs"; move_offline_into_place)
   deploy_docs
 fi
+
+# Zip docs
+cd "$FLUTTER_ROOT/dev/docs"
+zip api_docs.zip doc