It seems that if you want to play COH, your new computer can't be new-new after all:
So anything that runs Windows 10 is right out - the machine will have to run Windows 7 or 8.
I'm running Windows 10, and use a GeForce in my desktop and laptop. Both run COH well. (I'm sure AMD is just as successful.) So taking that literally, ANYTHING that runs Windows 10 being right out, isn't true. Microsoft not supporting something doesn't mean it won't work. OpenGL Support Forums confirms this; their tech support on problems are focused on Graphics Cards having compatibility, rather than the operating system.
Anything newer that runs Intel GMA, perhaps. Older systems with Intel GMA 4000 I'm able to get COH running on, so there may be a 'sweet spot' between that point and when Intel decided to drop OpenGL support natively. As a reality check, bring a USB with Tequila pre-downloaded with you. If it will run Paragon Chat, the game should be fine. (And mind the gap between "it runs", and "it runs well". I've never see Intel GMA deliver FPS higher than 60 without pointing your camera so that the sky is out of view on City of Heroes.)
Laptops, regardless if they're brand new or refurbished, expect to spend at least $600 to find a system with a discrete graphics card (or integrated graphics that aren't terrible, like Radeon Vega.) Most are going to be closer to $1,000, however.