Hi there. You have a few options here.
For access to Tweets older than a week or so (provided by the public Search API) you can use our commercial Gnip service. Unfortunately the search API itself has never provided data beyond that limited date range so there’s no way for us to grant any kind of additional elevated access on that.
For ongoing access to the Streaming API you can file a request at https://support.twitter.com/forms/platform for Elevated Access - these are typically granted on a time-limited basis for specific applications. Obviously this would only enable you to listen to the stream going forward rather than searching backwards.
The final option is our Data Grants programme, but this is currently closed to new requests.