Omeka is a flexibe open source webpublishing software that allows you to share born-digital and digitised content and metadata on the web. It’s not meant for managing or archiving data. Other tools - such as CollectiveAccess or Rosetta - are more suitable for this purpose. Of course connections between different systems can be made using web based technologies such as APIs and URIs. The metadata described in your research database can thus be aggregated and published on Omeka, providing a public access point to (the part of) the data you want to share.
Omeka is offered as a standard package or can be customised according to specific requirements, including design and development of special features that improve access to your data.