Nothing wrong with a 35mm camera for this.
It depends what you mean by vintage. There are lots of ways, and many things that involve the darkroom. But, if you don't have a vintage camera with a vintage lens and vintage film, one way is to use faster film (for the grainier look), and somekind of softening and/or diffusion filter. Basically, the idea is to make it look less sharp. You can take a cheap UV filter and spray it lightly with hairspray. Then, if it's black & white, have it printed with a slight sepia tone. You can make a 35mm picture look like it was taken in 1901 with this technique. Another cheap way is to buy yourself a Holga, some Tri-X or HP5, and just snap away. The pictures come out already with a vintage look to them. You can also get an actual vintage camera. Those with 3 element lenses (like Novar, Apotar, Agnar, etc.) on vintage german 35mm and medium format folders) have the sharpness in the middle and more softness towards the edges, unless you stop them down too much (in which case they can be virtually as sharp as anything made today).