Is there a Jenkins plugin for downloading source code?
Perhaps you are looking for the Job Cacher plugin?
To quote the plugin page:

Summary
Enables caching of files on transient executors for use on subsequent builds to improve performance
Features

Item storage extension point supporting on master storage and AWS S3
Cache Wrapper for free style jobs that manages the cache
Arbitrary File Cache where user specifies paths to be cached
UI on Job page to review the job's caches
Supports Pipeline jobs with a cache directive
Cache Extension Point for other plugins to provide opinionated caching capability such as Gradle caches
Incremental storing of cache into storage to minimize data movement


However, these are for files created by the jobs, not needed by them. In order to use it in your case, you might need a few dummy jobs specifically for downloading the content from the arbitrary URLs, and then caching them in your Jenkins cache. I guess you'd have to write the script for that.
To be frank, I've never used this plugin, and what I describe above doesn't seem to be an elegant solution. Your question specifically ask for a Jenkins plugin, however I believe the general problem is how to cache job dependencies in Jenkins. If you agree to this wider interpretation, there are some other options.
I have often solved the problem you mention using a combination of managed scripts and persistent storage which Jenkins and workers have access to, in Jenkins.  If you need really arbitrary dependency management, a tool like Spack or EasyBuild may be useful. Although these have evolved in the HPC domain, they do quite well at caching dependencies for workloads.
However, since you mention just a few dependencies (specific sqlite versions e.g.), I think it may be worth your time to write a script which does the caching for you and, together with globally-available persistent storage, do the caching and retrieval that way.