Second Amendment supporters are often interested in the Fourth as well; I think you would find a very solid group of them (particularly those who concealed-carry, rather than just hunters) are strongly opposed to this sort of thing.
I'm not a conservative, but I can see the reasoning behind it: they know very well that this is one important step towards building a national database of personal information (or more likely, augmenting one that already exists). Actually, I have found more on the right concerned about privacy, government surveillance, etc, than on the left because this concern naturally meshes with the right's "small federal government" stance.
Remember how Social Security numbers were "just for Social Security -- not any kind of personal identifier"?
But perhaps the left, which has just as much ability to be hypocritical as the right, will see the light starting...oh, say, 12pm on January 20.