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Is there an open source alternative that could be self-hosted and configured to run automated and periodical checks?

While not a web or automated option - if you want to run a quick crawl and scan on your apps you could try OWASP ZAP, it also has quite a few handy plugins - https://www.owasp.org/index.php/OWASP_Zed_Attack_Proxy_Proje...

Metasploit? You don't even need to host it (why are we so obsessed with making everything a website?)

The key part is automation, not a website. Make it part of your ci/cd pipeline.

Metasploit isn't the best choice for webapps, you probably want nikto or similar. Here's the owasp list: https://www.owasp.org/index.php/Category:Vulnerability_Scann...