I know this to be a common problem w/sidelights. However, more contemporary sidelights are much thicker or are built double-paned. Unless your sidelights are made of glass that's NOT meant for that purpose, it's a little 'difficult' for me to understand why you're feeling so much cold air.
Are you SURE the air is radiating through the stained glass and not around the door, itself? I mean, there isn't much glass to sidelights. The effect you are experiencing isn't out of the ordinary, but you're making sound like the air is almost 'blowing in'. This type of thermal transfer is normal where there is a 'surface' (which is your glass) between a cold surface (outside) and a warm surface (inside). The 2 'elements' will try to equalize each other with the cold air moving towards the warm air (since cold air sinks), and the warm air moving towards the cold air (since warm air rises.)
As far as installing a storm door, sure, you could do that but I'm not aware of a single-swing storm door for an entry door w/sidelights where the storm door will 'enclose' the entire entryway. Yes, you can get a storm door for just the entry door but your problem INCLUDES the sidelights.
As an aside, we bought our home w/an entry door and sidelights. We demolished that in place of double-entry doors. We then installed double storm doors too. Aestetically, it looks nice. (We have full-oval, solid oak entry doors.) This does pose its own set of 'issues'. (I'll leave that alone unless you're contemplating this approach. I will assume not.) I'm only mentioning this because the only way we could get a storm door was to go w/double-entry doors. The storm doors are full-view doors so you can 'enjoy' the doors from the outside too.
As far as insulating your sidelights, assuming the entry door and sidelights aren't leaking, I wouldn't do it. You could start a condensation/mold/mildew situation. And besides, they'd look kinda funny. If these sidelights and the entry door are part of the home's character, beings that you own an 'old house', you may end up just consider it something you need to live with.
If you have more info to offer, perhaps I can say more to your questions. My best to ya and hope this helps.
Jay J -Moderator
PS: God Bless America!