I don't think there's any universal database containing ongoing, widely representative HI assay output. The Antigenic Cartography group has made available some historical datasets that were used in published articles, such as
* Influenza A/H3N2 data published in Smith et al. 2004. Science 305:371-376
* Data from H1N1pdm09 assays in 2009/2010 (click on the links for html/Excel/OpenDocument tables)
* Data published in Koel et al. 2013. Science 342:976-979
Trevor Bedford et al. make their large set of historical HI data described in Integrating influenza antigenic dynamics with molecular evolution available in DataDryad.