This seems unrealistic, because the topology on a manifold doesn't have anything to do with the properties its structure sheaf, but I figured I might as well ask. This wouldn't be the first time I was pleasantly surprised about something like this. If not, is there any sort of way to attack differential geometry with abstract nonsense?
Even though schemes have singularities, "it's better to work with a nice category of bad objects than a bad category of nice objects". Manifolds seem to be perfect illustration of this fact.
Edit: Apparently my question wasn't clear enough. The actual question here is if we can define manifolds entirely as "functors of points" like we can with schemes (sheaves on the affine zariski site). There is no fully categorical and algebraic description of the category of smooth manifolds. When I say a "comprehensive functor of points approach", I mean a fully categorical description of the category of smooth manifolds.
Best Answer
Here are two things that I think are relevant to the question.
First, I want to support Andrew's suggestion #5: synthetic differential geometry. This definitely constitutes a "yes" to your question
--- assuming the usual interpretation of "abstract nonsense". It's also a "yes" to your question
--- assuming that "it" is the category of manifolds and smooth maps. Indeed, you can make it a full subcategory.
Anders Kock has two nice books on synthetic differential geometry. There's also "A Primer of Infinitesimal Analysis" by John Bell, written for a much less sophisticated audience. And there's a brief chapter about it in Colin McLarty's book "Elementary Categories, Elementary Toposes", section 23.3 of which contains an outline of how to embed the category of manifolds into a Grothendieck topos.
Second, it's almost a categorical triviality that there is a full embedding of Mfd into the category Set${}^{U^{op}}$, where $U$ is the category of open subsets of Euclidean space and smooth embeddings between them.
The point is this: $U$ can be regarded as a subcategory of Mfd, and then every object of Mfd is a colimit of objects of $U$. This says, in casual language, that $U$ is a dense subcategory of Mfd. But by a standard result about density, this is equivalent to the statement that the canonical functor Mfd$\to$Set${}^{U^{op}}$ is full and faithful. So, Mfd is equivalent to a full subcategory of Set${}^{U^{op}}$.
There's a more relaxed explanation of that in section 10.2 of my book Higher Operads, Higher Categories, though I'm sure the observation isn't original to me.