Given the science as Kenji Alt explained it in his article, the key fact of the vodka is that it is 80 proof, or 40% alchohol.
So beer, which usually is on the order of 4 to 6 percent alcohol (yes, there are outliers) will not have the same level of inhibition of the gluten formation.
The vodka is not the only factor in the recipe leading to crispiness and good crust, so you will probably still get very good results with the beer, assuming you like the flavor profile it brings to the wings, but it won't be quite the same.
If you use another liquer that is on the order of 80 proof, again assuming you like the flavor profile, you should get quite similar results. This would include mixing grain alcohol with water or orange juice or whatever to the appropriate ratio.
Other than some white wine for cooking, though, I don't buy spirits so I don't what would be more economical--and that might vary from region to region.