He'd probably have a selection of different weapons depending on where he was going, and why. Anything that could pass by as local, just in case he couldn't get a hold of anything else himself, or from whatever support he may or may not have there
He may, or may not have a single weapon of choice that he would take everywhere with him. This would likely be a .22 pocket pistol with a threaded barrel. Maybe a .25, because of the more reliable centerfire cartridge, but always with a threaded barrel for a suppressor. ALWAYS. As opposed to the threads on the PPK vanishing whenever the suppressor's not actually attached. Probably a Beretta 21, since it fits the bill in either caliber, and stays true to the roots of the character.
A second choice for situations deemed more likely to involve shootouts (deemed by his SUPERIORS of course... we may know that EVERY assignment would involve a shootout, but his superiors would never think that), would probably be a Glock 19. Something with a rail, so you could get the laser aiming module on-screen during the shootout. I picked the Glock 19 because there are factory 33 round magazines available for it, but any subcompact or compact 9mm that can have truly high-cap (30 rounds) magazines as an option would work.
He would definitely personally own at least 1911 that was customized to his own preferences, but it would probably only be seen if the story developed to him being hunted down in his own stylish Floridian coastal residence. Maybe Texas, along the gulf of Mexico. No threaded barrel for the 1911(s). The last thing he'd need is for his name to be NFA-registered somewhere for owning a suppressor that the Company didn't issue him.
A small selection of common hunting rifles and shotguns in his home too, with maybe just one single EBR, which would naturally be an AR-15.
The biggest difference would be the weapons he gets off the ground, so to speak. While the British Bond might dive for the MP5, the American Bond might dive past it for the Mossberg 500. That Walther WA-2000 in the bad guy's secret bunker looks nice, but that M14 over there might just be the ticket to shoot out his outboard motor before he can escape in his boat.
Another thing is that the American Bond would probably be more inclined to carry a knife than his British counterpart. Nothing like a Rambo knife. I'd say something like a simple single-bladed Wenger Ranger. It's a 'swiss army knife', so it's still typical civilian-fare, it has a large blade, and Wenger is less pedestrian than Victorinox when it comes to brand, fitting with Bond's usual high-class image.