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Was the email about you only sent to you?
In that case, it appears that you may have a case for defamation. It appears that this website sends anonymous emails for people, and it claims to be governed under Swiss law. However, American courts would take jurisdiction because the emails were sent to you in the United States, and the effects of the emails take place here. Therefore, you can file a civil suit against the website to determine who sent the emails, but that would be costly.
You could also hire a private investigator who may be able to do this for you (also at a cost).
How much depends on the attorney you retain. Attorneys would charge hourly for this. A typical hourly rate is in the area of $225.00, and this suit would require quite a few hours. You would likely have to pay upfront with a retainer. Private investigators' charges vary greatly, and you would have to contact one to find out what the charges would be for this type of work.
I also cannot tell you what your chances would be, as that depends on a great number of variables.