When they take the old siding off there is no plywood sheathing under the siding. That was the installation method back in the day, but building practices have evolved a lot since then and this is your best opportunity to really improve the structural integrity and insulation of your 2nd story.
It is just 1/2" insulation board nailed to the studs and siding straight over that. The 1/2" or 3/4" foam insulation boards have an R value of 4-6, which doesn't meet current energy efficiency code requirements, especially if the existing insulation in your wall cavities is poor, which is probably the case because if the age of your home. Thicker foam boards (1"-2") are better, but still aren't adequate to sufficiently insulate the walls. Your neighbors had their vinyl hung directly to the wall studs with a single layer of foam because it costs less and they probably didn't know any better.
Do I need plywood or OSB sheathing before the siding? Yes, for a couple of reasons. Firstly, sheathing greatly improves the rigidity of the structure by tying the wall studs together into a unified structural element. Secondly, the sheathing provides a nice rigid surface to hold and protect the insulation in the wall cavity.
Here's what I recommend.
1) Assume that when you remove the existing siding, you will find that the old insulation (probably wool or equivalent) has settled in the cavities and that only the lower 1/2 of the wall cavity was effectively insulated. I'd remove the old insulation and replace with fiberglass batts. Assuming that your walls were framed with 2"x4", your will need to install R-13 batts. If your want a significantly higher R value, you can have the wall cavities foam sprayed, but that will cost 2x -3x the fiberglass. Before you install the fiberglass batts, you will want to apply a bead of insulating foam all along the interior perimeter of the wall cavity where the backside of the drywall meets the back edge of the wall studs. This will completely seal the cavity and eliminate air movement.
2) after insulating, install 1/2" sheathing. OSB is fine in this situation and is somewhat less expensive than builder grade plywood.
3) at this point and depending on your Rvalue insulation requirements, you could add foam insulation panels to boost the R value. Hit the below links to learn about foam panels and their available thicknesses and R values. The second linked article explains the benefits and methodology of layering sheathing a insulating panels. After you open the link, click "download as a PDF" or click on the 2 page image and the full article should open.
http://www.finehomebuilding.com/tool-guide/articles/insulation-rigid-foam.aspx
http://www.finehomebuilding.com/how-to/articles/save-energy-rigid-foam-insulation.aspx?ac=ts&ra=fp
4) you will also need to install a vapor barrier like Tyvek or tar felt paper. This is generally stapled directly to the Sheathing and I think it can go between the sheathing and the foam insulation if you go that route, but I'll need to go do some research on the proper layering of the vapor barrier relative to the foam.
5) install vinyl
