You are going to want to read the notes on here about how to post. Because you are posting in a homework help forum you need to follow certain rules.
A vortex is a solution to equations of motion. A Lagrangian is a way of expressing the equations of motion for a particular system. That is, you are putting things a bit out of order. A Lagrangian is not "for 2D vortices." It is the Lagrangian of a certain system, and some systems happen to have vortex solutions.
Maybe you meant: Can we name a system that exhibits vortices? And what is the Lagrangian for that system? And how can one calculate a vortex in that system?
First part is easy, if you count a smoke ring as a vortex.
For the rest, you should go back and ask your question correctly.