Some illegals may be here for good reasons, but some are not. You don't have to be a US citizen to obtain a gun permit in my state of Indiana. Aliens are allowed to carry, just not illegal aliens. That makes perfect sense.
My wife is not a US citizen, she is a Canadian citizen. In a few more years she will be a US citizen. But we have gone through all the legal hassles, and expense, that Uncle Sam requires so she can be here legally. And yes, she does have an Indiana LTCH. You would want to know someone's background before you entrust him with such a huge responsibility as carrying a lethal weapon. I don't want a felon carrying a handgun, weather they happen to be a US citizen, or an alien. I'm all for non citizens carrying as long as they are a law-abiding person of upright character. Illegal aliens, hell no - we know nothing about who they are or what they have done. Let them go through the process and obtain their visa, then go for it and get a permit to carry.