I also see your related post in the suggestions forum (and one in the unrelated UN topic). I will respond here.
Endnote has a number of record types that are designed to contain the information for web sites, unpublished works, personal communications etc. The output for these is not always implemented in the thousands of styles though, and I think that one can request that they be updated via the styles update pages at endnote.com on a style by style basis. But APA 6th does include these particular record type outputs in the style, for example.
It helps to know exactly what a specific publisher requires in those citations and bibliographic outputs though, and this is not always specified, thus sometimes the author is better positioned to define this. The power of endnote is that you are empowered to incorporate what ever you like via the edit styles facility, and you can post updated styles here too, helping the community. The problem with “power” is that the software out of the box isn’t always intuitive, but that is what these forums are for!
At times there are challenges where you would like a “short author” form in the citation and a longer form in the reference list at the end. this usually involves either editing the citation (hide author and put the short form in the prefix) or if a common need, using some of the tricks with custom fields explored in other threads (Here is a recent discussion to automate this).
Unfortunately, the solution that have provided includes much of workaround and manual steps and didn’t introduce a workflow to cite the web sites
It is quite easy to do the citation and referencing from the “Google scholar” since we have the feature “import into endnote” that imports the citation and referencing to the Endnote and then to the Word. (attacement 1)
Please, remember that we have been sturgling till the X5 version for simple issue which is most frequently needed, that is: to show the in-text citation as Author (year). That has been requiring us to do much of work manually. But now in the X5, it is a simple as that! (attachement 2)