If I understand the question correctly Stefan is asking for an Ext interpretation of the polydifferential Hochschild cochain complex. Elements of this are not just continuous linear maps $C^\infty(M)^{\otimes n} \to C^\infty(M)$, but they have to be polydifferential operators. (This version of Hochschild cohomology is used in Kontsevich's formality theorem).
Anyway, one can understand the polydifferential condition as follows. Consider the jet bundle $J$ on $M$; this is an infinite-rank vector bundle whose fibre at a point $p \in M$ is the algebra of formal power series at $p$. If we choose coordinates $x_1,\dots, x_n$ at $p$, then we can identify the fibre $J_p$ as $\mathbb{R}[[x_1,\dots,x_n]]$.
It's standard that $J$ is a left $D$-module. Further, the obvious product on the fibre of $J$ makes $J$ into a commutative algebra in the symmetric monoidal category of left $D$-modules.
Then, one can take Hochschild cochains of $J$ in the symmetric monoidal category of left $D$-modules.
This is the same as the complex of poly-differential Hochschild cochains. The key point is that $D$-module maps $J^{\otimes n} \to J$ are the same as polydifferential operators.
Of course, this means that you can apply any of the standard interpretations of Hochschild cohomology in this context (e.g. $\operatorname{Ext}_{J \otimes J}(J,J)$).
One needs a little care with these definitions, because $J$ is a topological $D$-module. However, if you take continuous $D$-module maps and appropriately completed tensor products you get the right answer.
Best Answer
maybe these notes by Vezzosi can be helpful for some
http://www.dma.unifi.it/~vezzosi/papers/derivedintctgtcplx.pdf