I know that you cannot apply color ramps to the outline of a polygon, so essentially what I want to do is add a gradient fill with the inner color of the polygon being "no color", and then apply a color ramp to the outer color of the gradient fill.
This would help achieve classification of the polygons while still allowing the interior of the shape to be transparent.
Is there any way to achieve this by creating a custom style?
You can change the fill type of the polygon to by Gradient Fill Symbol in the Symbol Property Editor when you click on the Edit Symbol... button in the Symbol Selector window. Change the Style of the gradient from Linear to Buffered and it should do what you are asking.
From there you can change the Colour Ramp to be no colour to whatever colour you wish. If you lower the percentage of the polygon that is covered by the gradient to 5 or 10%, it will only show on the outer part of the area. Adding more intervals, from 5 to 25 or so will make the gradient appear more smooth.
It is probably necessary to up the transparency of the layer to make it look good.
Be warned - adding this type of style to a layer will definitely slow down your computer.
You can create buffers of negative values, then color those buffer features, while maintaining no color for your original polygon
