How can I remove the white background on this logo?

I'm not quite sure what you mean by "solidify…into a pdf" since you've already created a pdf.

To remove the white fill from the hanger, select the hanger with the Selection Tool (black arrowhead), then go to the Tools panel, Control panel, or Properties panel and find the controls for fill and stroke. Click on the fill button, which will show in the Tools panel as a large white square, then on the no-fill button (square with the red slash). The hanger should now be just the blue stroke. There is no other fill or background.

Your document also has several stray paths and type objects that you should delete: go to the Window menu > Layers to open the Layers panel. Click the black disclosure arrow to the left of Layer 1 to show the list of objects in the layer. Click to the right of the targeting circles at the right of the panel to select items to delete (shift-click for multiple items), then press delete. Obviously, do not delete the items "David's", "Thrifts", and the path that is the hanger.

Peter