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I am working on a PCB for a sold state buffer circuit that uses transformers on the input and output. I am wondering if anyone can offer any insight into filling in the PCB spaces between the traces with copper that is connected to chassis ground.

Hi,

I logged in Later, saw your post,
I suppose you are using Eagle for design.
what you want to do..

have a PAD to connect to chassis ?

and one more thing, the gap between trace and Ground Plane is very less, increase them

In Eagle -

Use the POLYGON tool to draw the area that you want filled, there are many options that define how the polygon behaves and it is easier to get them right before you put it in, but the CHANGE button will let you change any of these attributes later on
If you want that to connect to ground, NAME the polygon to your ground net name (GND, AGND, etc.) you can connect these polygons to any net you want

Also, in Eagle, the ISOLATE command changes the spacing of the polygon to adjacent traces.