I'd like you to ask if it's ok to use the word "coon" as part of a company name? The website isn't related to racoons at all, but has a racoon head in the logo. Will it offend visitors? As a foreigner I don't get all the finer points of the English language.
Probably the most sensible thing to do in this situation if you want a racoon-themed corporate identity would be to keep the "ra" in "racoon".
The word "coon" is in fact highly offensive to black folks in the USA, and there are frankly a lot of people here who have never seen a real-live racoon and never heard the word "coon" used in any way other than an insult to black folks.
There are indeed other (non-black) folks, particularly those who enjoy hunting, who will say "coon" to refer to the animal and mean nothing racial by it at all. However, for general public consumption, it should really be avoided.
Despite the election of an African-American President, race continues to be a very sensitive issue in the US. History of slavery and ongoing discrimination against African-Americans and many other groups make the use of any term that is considered ethnically offensive or insensitive a source of concern.
There has even been a movement to sanitize Mark Twain's work to remove the use of the term we euphemistically call the N-word. Many in the literary community have resisted the change to an important historical work and point to both the usage at the time and Mr. Twain's overall thoughtful approach to racial issues.
There is even a current controversy concerning the long standing name of a professional football team, the Redskins. There are arguments on both sides, but clearly some people are offended and upset.
Even though the word coon has a somewhat common non-offensive meaning, some people (and not a negligible number) will take offense to a new, non-historic coinage.
If you do not want to offend, do not use it.
Growing up, I always heard the animal referred to as a "raccoon." I believe "coon" was used almost exclusively as a racial slur. But then, my childhood was pretty suburban. I can imagine I might use the word "coon" if I ever have occasion to discuss the hunting of raccoons ("That's one heck of a coon dog.") or if I were asked to perform a condescending impression of a country bumpkin ("Mama's cookin' coon for supper!"), but otherwise I don't think I'd normally use coon in place of raccoon. That said, combining "coon" as a suffix with another term, as you are proposing, may remove the negative connotation. At least I've never heard the racial slur used that way. To me, GadgetCoon sounds like a subspecies of raccoon.
If you have a racoon logo I don't think it would be inherently offensive to anyone. Perhaps write it as 'coon in the logo with the apostrophe helping to indicate that it's short for racoon. I don't know if you can tell us the exact name, but if we know the whole context it might help to be more certain about our advice.
When it comes to offensive or near-offensive names, it seems that we have become a bunch of hypersensitive neurotics.
GadgetCoon with a racoon logo would seem to me at least to be perfectly OK, but I am not a black person.
We have the big companies SurveyMonkey and MailChimp, and both chimp and monkey have been used in the past as pejoratives directed at persons of the black race in the USA. These two companies have exactly as much to be with chimps and monkeys as GadgetCoon has to do with racoons, but I have never heard of anyone getting riled up at them for their names.
I suggest that GadgetCoon is probably OK as a company name.
Edited to Add: I consulted with my black friend who is from the South, and he couldn't see anything wrong with it, though he thought there were those who might.
I suggest that a racoon logo is pretty cool, and there doesn't seem to be a particular reason to shorten "Racoon" to "Coon". In my personal opinion, "GadgetRacoon" sounds better than "GadgetCoon" anyway, and doesn't punch anyone's buttons.
ETA: would the downvoter care to comment about his/her downvote?
If your website has nothing to do with raccoons, why have a raccoon head as a logo or even on the site? Unless the company founder's name is "Coon", why encourage problems with no good cause to do so? The easiest answer would be avoid the problem altogether and get rid of the raccoon and find a better company name.