In principle, yes, you can compute all chemical reactions by Feynman diagrams, since the underlying theory is QED.

In practice, to a reasonable accuracy, the mechanism of a chemical reaction can be describe by the transition state theory, ref. Atkins, "Physical Chemistry". The theory uses many assumptions of the transition state related to the equilibrium state statistical mechanics and usually it works.

Along this line, we may use density operator/matrix, than a single state vector to describe a reaction. It is physically plausible since there are $10^{23}$ molecules in a realistic scale.

Still, it is possible to use path integral with statistical mechanics, maybe wiki and references therein
http://en.wikipedia.org/wiki/Path_integral_molecular_dynamics

There is an underlying issue in both transition state theory and path integral molecular dynamics, namely constructing the potential energy surface (defined by the Born-Oppenheimer approximation). There are some techniques with Feynmann diagrams as well, ref. Rev. Mod. Phys. 79, 291–352 (2007) .