Not sure it is relevant to your problem, but how are "[downloading] an image of a URL" and "[creating] a local shortcut to a different URL" examples of using SolidWorks custom properties?
Anyway, if you want a list of key-value pairs (for example, a custom property name and its value) in VBA then you can use a Dictionary object. I personally have not taken advantage of this data structure enough; I tend to use the Collection object, which is very similar to a List<T> in .NET or Java. For example, in a Collection I might have all of the odd items be the custom property names and the even items be the custom property values.
There's a Collection example in my Macro Library under the VBA section. The Dictionary object works similarly, but you can easily google examples if necessary.
The URLs are being stored in the custom properties, and VBA is going to be taking those and doing things with them.
The dictionary data type looks promising, and the collections data type might work too. I'll have to look into it to figure out which is best suited for what I need to do.