Добавил установку crontab и запуск задания на обновление сертификатов и ключей.

Dockerfile

@@ -2,10 +2,11 @@ FROM centos:7.4.1708
 COPY nginx.repo /etc/yum.repos.d/nginx.repo
 COPY generate.sh \
      renewal.sh \
+     docker-entrypoint.sh \
      /tmp
 
 RUN yum install -y epel-release \
-    && yum install -y yum-utils python2-certbot-nginx openssl ca-certificates mailx nginx \
+    && yum install -y yum-utils python2-certbot-nginx openssl ca-certificates cronie mailx nginx \
     && yum install -y https://centos7.iuscommunity.org/ius-release.rpm \
     && yum install -y git2u \
     && yum clean all \
@@ -15,12 +16,14 @@ RUN yum install -y epel-release \
     && touch /var/log/letsencrypt/letsencrypt.log \
     && chmod +x /tmp/generate.sh \
     && chmod +x /tmp/renewal.sh \
+    && chmod +x /tmp/docker-entrypoint.sh \
     && ln -sf /dev/stdout /var/log/letsencrypt/letsencrypt.log \
     && ln -sf /dev/stdout /var/log/nginx/access.log \
     && ln -sf /dev/stderr /var/log/nginx/error.log
 
 COPY default.conf /etc/nginx/conf.d/
 
+ENTRYPOINT ["/tmp/docker-entrypoint.sh"]
 
 EXPOSE 80
 CMD ["nginx", "-g", "daemon off;"]

docker-entrypoint.sh

@@ -0,0 +1,7 @@
+#!/usr/bin/env bash
+
+# Запускаем crontab и добавляем задание по обновлению сертификатов и ключей
+/usr/sbin/cron
+echo "0 0 * * * /tmp/renewal.sh" | crontab -
+
+exec "$@"