What you need is someone that does "tap removal". If there are machine shops in your area, they will know who has the specialized equipment to do this. Or look in the yellow pages under "Machine Shops".
Basically, the bolt/tap is electrically vaporized with a pointed probe.
In our area, North San Diego County in Southern California, I have used Slivnik Machine in San Marcos. They have removed several dropout screws as well as taps from dropouts for me. You need to tell the machinist the tap size, they generally aren't familiar with bike parts...it almost always 3mm x .8mm.
It is pretty amazing that this can be done without harming the paint, but it is possible. I believe that last time they charged around $50 ... that's much better than replacing the dropout and repainting/rechroming.
They may be the same company that Jimbo at Cyclart used, as mentioned in the second reply below.
If you're really cheap, I once saw an article in a magazine on how to build a simply EDM to do this. Required about $20 in parts... a light bulb and a diode were the major items. No, I don't remember the exact magazine, but it was probably Popular Electronics, Popular Science, one of those types.
--- snip --- Here's a plea for help going out to the frame guys: I broke off a dropout screw just as the end was reaching the dropout. Now the bad news: the screw broke off nearly flush on the other side. What do I do now? Drill and E-Z out? Any recommendations? Thanks.
Breakin' 'em off in Freedom, CA
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Jody
When I had that happen, I shipped the frame to Cyclart. They have access to a machine shop-type tool which will do the job with no trauma to the frame. Check with Jim.
Ken, near St Louis-- hot
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