Yes, It is possible to formulate the EL relation for 6-DOFs robotic arm. However, to avoid computational cost and time, researchers used software based approaches such as Matlab/Simulink, Matlab Coding with Symbols, Peter Corke Robotics Tool Box etc.
Moreover, unless your system is a hybrid one, i.e., combination of serial and parallel mechanisms, finding EL equations are straight-forward.
Yes, It is possible to derive dynamic equations of a 6-DOFs robotic arm using Euler-Lagrangian mechanics. To start with, you can refer to Chapter 6 of the book titled "Introduction to Robotics" by J. J. Craig, where you can find the example problem for finding the dynamic equations of RP Manipular using Lagrangian formulation.