The suggestion you gave is the one given by Google Translate, and it does make very little sense.
Google Translate is very unreliable with Latin, and should as a matter of principle never be trusted.
A much better translation is often obtained by looking up the words one by one in any online Latin dictionary.
If it is anything important, please always check the translation with someone who knows Latin.
The first and most important step is to find the correct words.
Here is a suggestion:
- fellowship: A good word for a fellowship, association, union, community, or society is societas. The word communio does not seem to refer to an organization, at least not as clearly as societas.
- chiropractic medicine: A good general word for medicine is ars medicina or simply medicina. To specify that it's the chiropractical kind of medicine, an adjective is needed. Although it is almost certainly not classical Latin, the (originally Greek) adjective chiropractica seems appropriate.
- sincere: Perhaps fidelis would be appropriate? It means trusty, sincere, or faithful. There is also sincerus, but it doesn't seem to be very close to the English "sincere".
Putting all of this together with the correct forms gives:
sincere fellowship in chiropractic medicine
societas fidelis in medicina chiropractica
If you want to drop the word "medicine", you can simply drop the word medicina.
The implicit ars goes well with the plain chiropractica as well.