Every FHC has a page on the FamilySearch research wiki. For example, here's an FHC in my area:
Sandy Utah Crescent Family History Center. The FHC director (or any other responsible contributor, for that matter) can edit that page to make it reflect the current details for the FHC.
Many of the current FHC pages on the wiki are only stubs (the one I linked to is an example of that), but others have quite a bit of information (such as the
Riverton FamilySearch Library). The wiki format allows for multiple pages for a particular FHC, if you want to put that kind of effort into it.
As for creating a separate site, you should review the Handbook 2,
Section 21.1.22, which covers the creation of websites. The policy is rather clear for stakes, ward, temples, missions, and visitors centers, but Family History Centers are not specifically mentioned.
Questions that can benefit the larger community should be asked in a public forum, not a private message.