You don't neede vintage gear to get that kind of sound, but you have to think about the kind of stuff that they had.
For 50's and early 60's sounds:
i.e., For the toms use thin coated heads top and bottom, and tune them up high.
They typically used one, and sometimes 2 mic's to mic the drumkit (a room mic, and maybe a kick mic - but the kick would have been mic'ed out in front of the kick perhaps with a bunch of blamkets touching the solid front head.
For late 60's and early 70's - the Evans Oilhead is a definite need.
Why? Because they would cover the top heads with duct tape.
Listen to those tracks - they have no attack on the toms. A regular modern head will give you a lot of attack - so you either have to use an evans Oilhead - which has less attack, or cover the heads with duct tape, or a towel.
Again - a lot of these recordings were made with a room mic - NOT an overhead. They kept some distance between the mic's and the drums back then.