Is there a way to trigger a build when a SVN External is updated?

I currently have a project which includes a ton of externals which needs to be rebuilt as the externals are updated, to keep the artifacts fresh. I cannot figure out a way to trigger a svn build when an external changes, only when a change is made in the branch itself.

Is there some feature of TeamCity that will do this, and if not, what is the best way to add this build trigger?

You can choose the External support you want under the VCS root settings:

If you are asking for support to detect if the external change happened in the context of a particular branch / location where it is added as an external, and ignore the changes when the external gets changed in other location, no, that is not possible and, frankly, not how externals are to be treated.