$C^\infty(E)$ is a Frechet VECTOR space. Thus its tangent space at each point equals $C^\infty(E)$ via its affine structure.

# Added: 

This is also true if $M$ is not compact. However, for a fiber bundle $Q\to M$ one has to be more careful with the topology (if $M$ is not compact). See 10.10  of 

- Peter W. Michor: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Shiva Publ., Orpington, (1980) [(pdf)][1]

for an answer. In principle, the tangent space is the space of sections of the vertical bundle of $Q$ restricted to the the image of a section.
There is also Sections 42, 43, .. of 

- Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 [(pdf)][2]

where a more easily handable notion of differentiability is developped and then used.

These methods also work for Sobolev spaces of sections. See 

- MR3135704 Reviewed 
Inci, H.; Kappeler, T.; Topalov, P.
On the regularity of the composition of diffeomorphisms. (English summary) 
Mem. Amer. Math. Soc. 226 (2013), no. 1062, vi+60 pp. 


for the basics of these.


  [1]: http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf
  [2]: http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf