Probably not. There are several reasons.
First, Tolkien never names them as such. In fact, there's only that one passage in the entire work that even suggests that foxes are intelligent, and it only suggests it -- it doesn't say so. (Personally, when I read it I took this to be anthropomorphizing an animal's reactions so that we could understand it. And, doubtless because it better flowed with the story than a more scientifically or philosophically correct explanation.)
Secondly, Tolkien is not writing a D&D rulebook, so, while he had thought about these things in great depth, he does not write everything he knows in the books. Nonetheless, the general categories of beings seem to be:
Ordinary mortals: Men, Hobbits. They live; they die; they leave the world and will definitely sing in the final Music of the Ainur.
Elves: They are bound to the world and will live in it until its end. Their bodies can die, and they can spend a long time in Mandos, but eventually they will be reborn in some fashion. Wash, rinse, repeat. After the end of the world, they will probably also join in the Music.
Dwarves: They live once (modulo Durin), go to Mandos, and then? The Elves don't know. The Valar haven't said (if even they know). But the Dwarves believe that they, too, will sing in the Music. So they're basically another race of mortals.
Ents, Eagles: Apparently these are plants and animals inhabited by spirits called by Yavanna and Manwe, respectively. It's unclear what the spirits are -- maybe very junior Ainur? -- and hence it's unclear what happens to them after death. But it is clear that they are special exceptions to the natural world, and this is shown by their forms: Walking trees (more-or-less -- not like anything else, anyway) and giant birds that can carry a full-grown human. To count foxes in this category, I think you need either a canonical statement or, at least, that the foxes be distinctly different than the ordinary fox-in-the-chicken-coop kind. We have neither.
Orcs, Trolls, Dragons, Wargs -- these are apparently all natural people or beasts corrupted (of "made in mockery") by Morgoth to the point where they are separate species. There is no information on what happens to them after death.
Ainur (Valar, Maiar, Balrogs): Like the Elves, they are bound to the world and will stay there until the end of the world when they presumably will rejoin the Ainur in the Music. They can't be killed, but they can be diminished by the consequences of their own evil actions. In their natural state, they can embody themselves as readily as humans change their clothes, but if they diminish themselves through evil that ability can diminish or be lost entirely. There seems to be no end to the extent they can diminish themselves, but there is no reason to think they can end themselves entirely.
Now we have foxes. As far as we can tell, they are mortal like humans, but there is no evidence that they will join the final Music nor that one of the Valar made them by calling spirits to inhabit them. (Either could be, of course, but I don't believe there is anything in the Canon to support the idea.)
Finally -- though it's only Entish lore -- there's:
Learn now the lore of Living Creatures!
First name the four, the free peoples:
Eldest of all, the elf-children;
Dwarf the delver, dark are his houses;
Ent the earthborn, old as mountains;
Man the mortal, master of horses:
This suggests that foxes were not widely held to be one of the "free peoples".