We are publishing a map for use in several EC countries.  The overall
language of the website is in English.  When we use place names on the
map which language Vernacular should they been in?  We don't wish to
offend our customers in France for instance by using the English or
German spelling of place names if that isn't common practice.  What is
the precedent in situations like this?

Hello faroff:

Thanks for the fascinating question. As a map-collector and long-time
National Gepgraphic member, this question hits really close to home.
:-)

While I was not able to find one definitive "rule" or "standard" for
what you are asking, what I am going to suggest to you revolves around
what National Geographic does in this situation.

[Taken from the Europe map, page 53, of the "National Geographic Atlas
of the World - Revised Sixth Edition.]

* For country names, use the English.
  * e.g., France, Germany, Russia, etc.
* For city names, use the local dominant language of the country that
city is in
  * for "larger" cities, also place the English translation
beside/under the name in parenthases (unless that translation is
identical to original)
    * e.g., Roma (Rome), Munchen (Munich), Baile Atha Cliath (Dublin)
  * if the local language uses a different alphabet, then use the
Roman alphabet translation of the city name in the local language
    * e.g., Kyyiv (Kiev), Warszawa (Warsaw), Athinai (Athens)
  * use accents where applicable and where possible with your
character set

Following the above rules should keep you from offending anyone. :-) 

Since you are making a website here, you also have the opportunity to
create a click-to English-only version of the map. However, I'd make
sure that's not your default. If you want to go way out on a limb, you
could also make versions in other languages, such as French, German,
Spanish - with just that one language in them. But that's probably
more than you want to undertake.

I hope this information helps with your research.                    
                    
If you need any clarification of the information I have provided,
please ask using the clarification feature and provide me with
additional details as to what you are looking for. As well, please
allow me to provide you with clarification(s) *before* you rate this
answer.
                    
Thank you.                     
                    
websearcher-ga                     
                    
                    
Search Strategy (on Google):                     

"place names" Turin torino
toponymy "european maps"
toponymy rules OR standards

Exactly what I would've said ;)

Great answer.
An additional thought; one could also provide a localized version
depending on the visitor IP (that is, automatically detecting the
country). It's not fail-safe, since not everybody accessing the site
from e.g. Germany must be German, but it's quite pragmatic.

Thanks. I appreciate the advice.