Okay so here's the thing. Economists love math, but good economists recognize that they are doing a social science. They don't do math for the sake of doing math. 

Math is a tool/technology for explaining an economic system. Can we endow an economic system with a geometry (symplectic or otherwise)? Of course we can. But what does it buy us in terms of the social theory? If the answer is "well it's neat", you are going to be hard pressed to find an economist that gets excited for it. Economists have incorporated some aspects of topology and set theory into their economic theory, not because "math is fun", but because they could prove theorems of stability using fixed point theorems.

For a geometric interpretation of economics to be successful, and it could be, it needs to provide tools that prove something about economic theory that is unproven, or that leads to greater insight into human behavior. 

Symplectic geometry falls in this grey zone. It has just enough interesting or novel to draw an ambitious young economist in, but (so far) not enough to get economists in general excited about it being incorporated into canonical economic theory. It just doesn't help us understand something new, just the same old stuff at a slightly deeper level. 

Now, will someone come along and pick up symplectic geometry and prove something important in economic theory?? Who knows? We had Kakutani's fixed point theorem lying around and economists largely ignored it,  preferring Brower's fixed point theorem. The conventional thinking at the time was that economics deals with functions, not correspondences. Kakutani's fixed point theorem was a mathematical toy. And then along came John Nash. He reinterpreted game theory to include mixed strategy solutions as a correspondence for the canonical forms of game theory, and oh hey look, I can prove it is a stable solution using Kakutani's fixed point theorem. Is an important idea in economics waiting to be solved using symplectic geometry or will it remain a toy mathematical structure? I don't know. Maybe the right question hasn't been asked yet.But if someone finds "that question" or "that application", they are virtually guaranteed a nobel prize.

Tl;dr
Certainly economists are aware of symplectic geometry, they just haven't figured out why they should use it, yet.