All documents should simply be dumped into a common repository. S3 would do fine. Define some standards for common document metadata so each document is identified by case, author, type, etc. Enable S3 version histories.
Then create an API for creating cases and uploading documents. This will require some controls, logins, security, and some facility for billing users for filing fees. If the court just provides an API, they can stop there.
Private companies who want to provide a user interface to lawyers and the public can do so. I'm sure that more than a few will make searches and document downloads free.
And that's it. If the government would just get out of the way, this could be done in six months to a year and would be very, very cheap.