Fix submodule handling in dockerized tests
diff --git a/tools/run_tests/dockerize/docker_run.sh b/tools/run_tests/dockerize/docker_run.sh
index b19cc25..e525019 100755
--- a/tools/run_tests/dockerize/docker_run.sh
+++ b/tools/run_tests/dockerize/docker_run.sh
@@ -26,6 +26,7 @@
   # TODO: figure out a way to eliminate this following shellcheck suppressions
   # shellcheck disable=SC2016,SC1004
   (cd "${EXTERNAL_GIT_ROOT}" && git submodule foreach 'git clone ${EXTERNAL_GIT_ROOT}/${name} /var/local/git/grpc/${name}')
+  (cd /var/local/git/grpc && git submodule init)
 else
   mkdir -p "/var/local/git/grpc/$RELATIVE_COPY_PATH"
   cp -r "$EXTERNAL_GIT_ROOT/$RELATIVE_COPY_PATH"/* "/var/local/git/grpc/$RELATIVE_COPY_PATH"
diff --git a/tools/run_tests/dockerize/docker_run_tests.sh b/tools/run_tests/dockerize/docker_run_tests.sh
index 4e4ca18..c41734c 100755
--- a/tools/run_tests/dockerize/docker_run_tests.sh
+++ b/tools/run_tests/dockerize/docker_run_tests.sh
@@ -26,8 +26,9 @@
 git clone /var/local/jenkins/grpc /var/local/git/grpc
 # clone gRPC submodules, use data from locally cloned submodules where possible
 # TODO: figure out a way to eliminate this shellcheck suppression:
-# shellcheck disable=SC2016,SC1004
-(cd /var/local/jenkins/grpc/ && git submodule foreach 'git clone /var/local/jenkins/grpc/{$name} /var/local/jenkins/grpc/${name}')
+# shellcheck disable=SC2016
+(cd /var/local/jenkins/grpc/ && git submodule foreach 'git clone /var/local/jenkins/grpc/${name} /var/local/git/grpc/${name}')
+(cd /var/local/git/grpc/ && git submodule init)
 
 mkdir -p reports