Except that (at least some) Republicans respect federalism. Let each state decide for itself what gun laws it wants, consistent with the Second Amendment. Sounds conservative to me.
Unfortunately, that gets into a whole 'nother snarl over interpretation of the RKBA. What New York thinks vs North Dakota are different critters.
Another serious problem is that some States don't require any permit to legally carry concealed. There's nothing to have reciprocity with.
I like the fundamental idea that if you are considered by your home state to be a rational and moral enough person to be carrying the means to dispense lethal force, that recognition should not end at the state line. You, as a person have neither changed in demeanor (well, except for some tourists I saw in my youth) nor proficiency, and should not be summarily criminalized for a mere change in location across some line which cannot generally be seen, and exists only on a map.
If Police, Peace Officers and members of the military can have that acknowledged, then surely those private citizens who have gone through the training, background checks, and proficiency tests can be similarly treated.