G'day goings
Welcome to MusicBee and the forum.
Is MusicBee storing information locally on my internal hard drive?
Yes. By default the MusicBee(MB) library, (which is actually the MB database, not necessarily the actual music tracks), is located in your "My Music" folder.
You can change this in Preferences> Library> music library> Move Library
By default, MB also stores all it's settings and caches in %USER_NAME%\AppData\Roaming\MusicBee. A large, well-tagged collection uses up a lot of space for caching Artwork, Artist pics, etc.
If you use the portable version, with MB installed on the external drive, these AppData files will be placed there instead of on the C: drive.