I haven't tried this, but from the instructions (here: http://22.214.171.124/variantdb/index.php?page=tutorial&topic=newdata) my guess is that the datasets to export must have history sharing enabled when using their Galaxy wrapped tool. You could test out just sharing by URL and if that doesn't work, by fully publishing the data.
The first might be enough from the context of how the two servers interact. The documentation states the match is based on username, but the code from the export tool itself seems to be using the registered email address. These are two different account attributes in Galaxy (not sure about the other service). I would try using the same username/email with both (if available) but especially the same email if testing this out myself.
Note that to use the data transfer tool mentioned in the documentation link (above) assumes that their transfer tool (wrapped for Galaxy) is installed or can be installed on the Galaxy server with your data and that the server is public or maybe on the same network as the other database. This tool is not installed at Galaxy Main http://usegalaxy.org and I do not know if that is something we may do in the future. If attempting to load data from Galaxy Main without using that tool, then the dataset download URL might work, if they accept that sort of data transfer. Dataset download links do not require history sharing to be enabled.
How to share in Galaxy: https://galaxyproject.org/learn/share/
If none of this works, or you want the actual best practice usage from the other server, getting clarification from VariantDB, as Martin suggests, is the way to find out the exact requirements.
Hope this helps! If you do confirm how to do this, writing back with that information might be helpful for others.
Jen, Galaxy team