To make things coordinate-free, it is sufficient to reformulate differential equations in the form that makes use of exterior derivatives and exterior products of differential forms. Any set of differential equations can be cast into this form, the only subtlety being that it may require an infinite collection of differential forms to be introduced.
As topologist you may be aware of Sullivan's work "Infinitesimal computations in topology"
in which a sort of such equations were studied. Though to do real PDE it is necessary to work with zero-degree forms too, which he did not.
Such equations have applications in physics, e.g. you may write the equations describing black-hole without referring to any coordinates.

The typical system has the form $d W^A=F^A(W)$, where $W^A$ is a set of some forms valued in some linear spaces, $W^A$ do not necessary have the same degree, $F^A(W)$ expands only in terms of exterior products of $W^A$ with constant coefficients.

As an example, take one-forms $\Omega^I$, $F^I=f^I_{JK}\Omega^I\Omega^K$, then $d\Omega^I=f^I_{JK}\Omega^I\Omega^K$, the integrability for this equations implies $f^I_{JK}$ be the structure constants for some Lie algebra. The covariant constancy equations can be formulated in the same form.