And a blockchain helps to solve which part of the problem? Some were working on mirroring all data to a git repository. In theory, that allows for easy access on all the data, versioning (with commits) and - through forks and merge requests - collaboration and distribution. Also git is a distributed repository that clones the whole history to your local drive.
https://github.com/MITRE-Cyber-Security-CVE-Database/mitre-cve-database
But with the announcement of the cve foundation, I don't know whether they will really import all the data in this git repository.
You state that you did use the install script, but also that you want to run it with docker. Did you follow the instructions in their docker repository? It's quite easy to get it running - they included a complete docker-compose, a Caddyfile and all you need.
https://github.com/searxng/searxng-docker
Edit, I'm dumb, I misread.