Hi and welcome - I haven't grown a Rangoon Creeper for a long time and didn't see this problem on mine, but it was a huge rampant thing growing outdoors, so if it had I wouldn't have seen it anyway. It does look like some sort of soil-borne fungal problem or bacterial blight.
How big a plant is it, and how big is the pot? Have you re-potted it recently or in recent memory? If not, it may be worthwhile to slide the plant out of its pot and have a look at the roots. IF it's pot-bound, take it outside, hose off as much of the soil that's on it as you can, and re-pot in completely new soil. The infection, whether bacterial blight or fungal problem should be mostly eliminated by doing this. When it's in the new pot, prune off any stems that have the spotty leaves, right at their origin.
Did you put the pot outside for the summer? That's really the only way it would contract a blight like that as it's unlikely it would happen indoors.
I have experience with blights on my edibles, such as tomatoes and peppers here in Florida. Every fungus and disease not to mention bug lives here so we get it all. The first thing I always try when I see leaf anomalies that progress up the stems like that is a mild hydrogen peroxide douse. Dilute some H2O2 from the drugstore 1oz. to 32oz. of water and water the plant thoroughly with it. May not help, but it can't hurt. If you re-pot, I'd also do this peroxide douse, so the plant takes up the peroxide/water and hopefully it kills any blighters in the stems.
