Is it possible to get a setting from another extension?

I've developed a custom variable setter for selecting a particular build in artifactory. I would like to use the server, username and password from the Artifactory extension if it has been installed and configured, otherwise use my own settings.

Are there any buildmaster sdk calls I can make to get settings from another extension?

You could use the ExtensionConfiguration_GetConfiguration method, e.g.:

EXEC [dbo].[ExtensionConfiguration_GetConfiguration] 'Inedo.BuildMasterExtensions.Artifactory.ArtifactoryConfigurer,Artifactory', NULL, 'N'

and get the Extension_Configuration column, which contains the XML serialized properties of what you need - some simple XPath should be able to retrieve them.