Yes you can weld stainless and even exotic alloys with the stud weld process. However, for some applications, the capacitor discharge method is preferred over the drawn arc process.
When I was at GE we welded some jet engine parts that were Kovar and Hastelloy to tungsten. As you may be aware, tungsten melts at around 6000F and the Cr-Nickel alloys melt in the 3000F range. The CD process is so fast that the burst of energy does not care about the differences in melting temperature. However, the joint ends up looking more like a diffusion bond than a stud weld. And there are high speed automated CD stud welders that are like robotic machines. They can weld brass studs to very thin sheet metals in very very rapid succession. And now, there are even inverter power supplies with precise digital control and automated process charting, SPC, and bell curve tolerance based process controls and alarm systems.
We have a couple of these ( http://www.nelsonstud.com ) stud welding machines in our shop and we have had really good service from them...plus if we have any special applications come up on a job that we are not familiar with, they have good customer support. 
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