The tl;dr question is this: are economists using coordinate-free formulations in studying theory?
Borrowing from classical mechanics, the framework I have in mind for classical economics--involving maximizing principles--is the following. Let $Q$ be a smooth manifold parametrizing a `state space' of an economic system ($Q$ for Quantities, or stock variables). A utility function $S:Q\to\mathbb{R}$ induces a section $dS$ of $T^*Q$, whose image is a Lagrangian manifold $L$. When we choose coordinates $q^i$ on $Q$, the values of corresponding $p_i = \partial S/\partial q^i$ give the equilibrium or shadow prices. So "phase space" $T^*Q$ encodes both quantities and associate prices, and $L\subset T^*Q$ possible equilibria.
Conversely, let $L\subset T^*Q$ be Lagrangian so $\omega|_L=0$. Since cotangent spaces have a Liouville one-form with $d\theta = \omega$, the Poincaré Lemma implies that $\theta|_L$ is (locally) exact and hence there is a (utility) function $S$ so that $\theta|_L = dS$. Expressing the Liousville form in coordinates $\theta = p_i dq^i$ we recognize it as the key economic concept of income.
This establishes a neat relationship between two economic concepts of equilibrium: no arbitrage (no income on closed paths, $\int_{\partial \Omega} \theta = \int_\Omega d\theta = 0$ implies $\omega=0$) and maximizing utility (prices maximize utility or $\theta = dS$).
Furthermore, we can now apply Hamiltonian dynamics. Let me not go into this here, but just note that I have not encountered this formalism in any of the recent economics literature I have read (textbooks, articles or policy pieces).
I have written this up on SSRN but have not found much interest with economists I have approached. I have found a reference in Jan Tinbergen's PhD thesis (in Dutch). It discusses applications of variational calculus in physics, and economics in the appendix. Written in 1928, both economics and geometry have seen great development since--and I wonder if their paths have crossed more recently.
Update I should have mentioned Thomas Russell, who Carlo cites. His example is of a profit-maximizing firm, I am curious as to the more general case where the objective function is not explicitly known. The area condition for equilibrium he explains, was described by Samuelson in his Nobel lecture and inspired by reading Maxwell's work on thermodynamics!
Although I have labelled this as symplectic geometry, possibly contact geometry is more relevant. If the scale of the objective function is unimportant, and only relative prices matter, then the natural phase space is $\mathbb{P}(T^*Q)$ (as in thermodynamics).
Update 2 The question asks for economists, but Lee Smolin has written in the spirit using gauge theory on arXiv.
