Is there a coordinate-free definition of a differential operator?

There is a characterization due to Peetre which ensures that $P:\mathcal{D}(\Omega) \rightarrow \mathcal{D}'(\Omega)$ is a differential operator if, and only if $\hbox{supp }(Pu) \subset \hbox{supp } u$, where $\Omega$ is a open subset of $\mathbb{R}^n$. See: Peetre, J. Une Caractérisation Abstraite des Opérateurs Différentiels and Peetre, J. Réctification a L'article - Une Caractérisation Abstraite des Opérateurs Différentiels Théorème 2.

There are several generalizations of this result for manifolds and fiber bundles.

We can define them without explicit coordinates using smooth vector fields (sections of the tangent bundle). If $M$ is a manifold, then we can (a bit loosely) define differential operators of order, at most, $m$ to be finite linear combinations of, at most, $m$ vector fields.

If you want to be more explicit and careful on the construction and expression, first define $\text{Diff}^0(M)=C^\infty(M)$. Next, define $\text{Diff}^1(M)$ to be operators $P:C^\infty(M)\rightarrow C^\infty(M)$ of the form $P=V+f,$ where $V$ is a smooth vector field, and $f\in C^\infty(M).$ Finally, define $\text{Diff}^m(M)$ to be operators $P:C^\infty(M)\rightarrow C^\infty(M)$ of the form $$P=\sum\limits_{k=1}^K P_{k1}P_{k2}\cdots P_{kN_k},$$ where each $P_{kj}\in \text{Diff}^1(M)$, $K\in\mathbb{N}$, and $N_k\leq m.$