I am running 700x38 on my '73 Schwinn Sports Tourer. On skinny Wolber rims they measure out to 35mm, which is probably the max I'd go. 38 would fit but pretty close to the chainstays. I needed to use long reach Dia Compe centerpulls on the back to reach the rim. They work fine.
Fitting 38s usually means a 700C or 650B conversion on a 70s bike, with long reach brakes. If you don't want to use long reach brakes, a dedicated randonneur frame is your best bet. I will be building up a Japanese randonneur bike soon that was originally made to fit 650Ax42mm with cantilevers.