I'm a bit under 5'10",(a "bit" is a flexible term) with a 34 inch leg length and a 32 inch slacks inseam. I can ride bikes with traditional geometry anywhere in the size 56 range to size 62 range, with a size 57 to 60 being my ideal size.
So, at 5' 6", I'm thinking you ought to be on a bike in the size 52 to size 55 range. Go to your neighborhood bike shop and "stand over" a size 54 bike. If the top bar brushes your jeans, that's fine. If the top bar is pressing into your flesh, try the next size down.
You won't see many "vintage" road bikes smaller than about a size 54. Although most men in the 1965 to 1975 era were in the 5'6" to 5'9" height range, the most popular road bike size was "23 inches"...what is now sold as a size 58. And, in the more expensive road bikes, the second most popular size was "24 inches" and the third was "25 inches"...that is size 61 and size 63.
So, in 1965, a guy who was 5'6" was probably riding a size 58 bike...Yeah, it was probably "too tall" and his saddle was sitting about an inch off the top bar. But, lots of guys rode bikes with their saddle that low, and none are the worst for it, as far as I know.
So, you probably are going to have to settle for a size 21 (size 53) in a vintage bike...but I suspect that size may actually fit you rather well.