You won’t be able to run your Java program with the Wix platform. Depending on the size of that program, it might make sense to adapt it to expose an API over HTTP and host that on another platform, which you could then call from Wix. You seem to understand this but are reluctant to do it.
Depending on what the program does, you can rewrite it in JS and run that from Wix’s backend without calling any other API over HTTP. Maybe there is a gap in the documentation where it’s not explained where the backend actually is.
Wix does integrate Node into its technology stack but what is exposed to us as programmers is not pure Node. You can run JS only on Wix’s terms: there’s no file system access for instance. Node’s default classes are AFAIK also not exposed. You are supposed to use Wix’s own API and the npm packages they decide to add the library.
You can invoke your backend code from the frontend or schedule jobs to run periodically, but there is no option to have a process running constantly (though you can expose your own HTTP endpoints also).