I created a document using quick parts, but when I go to distribute it to my department, their systems don't have the quick parts in the documents building blocks. How can I move the quick parts with document. I am on 2007 now.

Check Help > "Quick Parts" in Word 2007 -- I suspect you will find what you need

I can't find anything in there other than that it is filed in a file called "Building Blocks.dotx" That file has nothing in it when I open it.

Create a template and store the Quick Parts in the template, not "Normal.dotx or Building Blocks.dotx". Then distribute the template to your users. They'll need to open a document based on it by clicking File > New > My Templates > YourTemplateName (not Ctrl + N).

I'm sure you can move the Quick Parts you've created by opening your template (click open, don't double-click -- that will open a document based on the template). Then go to Insert > Quick Parts > Building Blocks Organizer, locate your Quick Part, click Edit Properties, then select your template in the Save in box.

Building Blocks are stored in templates. By default, they are stored in one of the building blocks templates, but you can choose to store them elsewhere. As stated, you want to use a custom template for your document (not normal.dotm or one of the building blocks templates that come with Word). Distribute that template to your users. They do not open it, but rather use it to create a new document. It can be placed in their user templates folder or a workgroup templates folder.

Templates in Microsoft Word
Boilerplate in Microsoft Word

You may want to use Greg Maxey's Building Blocks utility.