Here is a sketch of one possibility how to prove this:
The first key step is to see that the orperator you are looking at has to be a differential operator. One usual way to ensure this is to require that it commutes with local differmorphisms (which include open embeddings). This implies that the operator is local and hence a differential operator by the Peetre theorem.
Having this at hand, the operator has to be induced by a vector bundle map $J^k\Lambda^pT^*M\to\Lambda^{p+1}T^*M$, where $J^k$ is the $k$th jet prolongation, and this bundle map has to be compatible with the natural action of all local diffeomorphisms. Such bundle maps are then determined by linear maps between the standard fibers of these bundles, which are equviariant for the actions of the so-called jet groups. (Formally, this is proved via higher order frame bundles.) In this case, the relevant group will be $G^k_n$, where $n$ is the dimension of $M$. This is an extension of $GL(n,\mathbb R)$, formally defined as the group of $k$-jets at $0\in\mathbb R^n$ of local diffeomorphisms of $\mathbb R^n$ fixing $0$.
Now on the target space $\Lambda^{p+1}\mathbb R^{n*}$, the jet group acts just via the usual action of $GL(n,\mathbb R)$ and this is an irreducible representation. On the other hand, restricted to the subgroup $GL(n,\mathbb R)$ the representation inducing $J^k\Lambda^pT^*M$ is the direct sum of the representations $S^{\ell}R^{n*}\otimes\Lambda^p\mathbb R^{n*}$ for $\ell=0,\dots,k$. Now there is only one summand which contains a copy of $\Lambda^{p+1}\mathbb R^{n*}$, namely the one for $\ell=1$. Hence your operator must be of first order, and its principal symbol must be the one of the exterior derivative (up to a constant multiple).
Hence you have a multiple of the exteiror derivative up to adding an operator of order zero, i.e. a bundle map $\Lambda^pT^*M\to\Lambda^{p+1}T^*M$ which commutes with the action of all diffeomorphisms. Representation theory of $GL(n,\mathbb R)$ immediately implies that such a bundle map does not exist.
There are lots of examples like this one discussed in the book by Kolar, Michor and Slovak (MR 94a:58004)