Are the current double hung windows double paned or insulated? What material are they made of?

If they are newer and insulated, you can create some issues of significance if you put on a storm window and can actually facilitate a window failure.

There are interior storm panels that can be fitted as well as interior magnetic storms that seem to help with drafts.

I agree. Interior storms are only viable option, especially if your windows have low-e. These coatings do a great job of reflecting heat to the point where they have melted siding on neighboring homes, plastic car accessories, etc. Just imagine what that much heat would do to your vinyl window frame if it were to build up in a space sealed by a storm.

The windows are vinyl and double paned. According to the certification tag they are NFRC 100 and dated 2002.
Are the inside storms custom-made or standard?
Thanks

Most inside storms are custom made.

Do your windows have Low-e in them? Do they have a gray'ish - green'ish tint to the glass.