Power options for inside cabinet drawer outlet
You don't do it with 120V unless you have AHJ approval for your installation.  I can't imagine that happening; there's some argument to be made in NEC 400.7 for flexible cords for equipment which moves, and maybe they'd approve it with armored cable or something... but instead I think the AHJ is going to take one look at this and say &quot;Heck no&quot;.
12 volts DC
So what we do is, use 12 volts DC. As long as the entire system is under 55W, we get to play under the easier Class II wiring rules.
So you have a UL-listed 12V DC power supply making up to 55W of 12V DC.
Then we come over to the drawer with, well, I would still use AC mains-grade armored cable, say 12/2, or a UL-listed pantograph arrangement as you see pictured in that product. It needs to be made to flex, though - you can't use in-wall wiring.
How do we get from 12V to USB?  LOL, that's easier to find than eggs.  Every convenience store, gas station, cell phone shop, big-box of any kind, etc. sells 12V USB adapters.  Usually right up by the checkout.
The USB charging hub may already do this for you.
Given the difficulty of getting a UL Listing for a 120V device, and the ease of simply including a commodity &quot;wall-wart&quot; power supply that is already UL-listed, I would expect most USB charging hubs already make a DC transition using a &quot;wall wart&quot; or &quot;power block&quot; of some kind.
They don't even make the wall-wart; they buy it from someone else who already slogged through the difficult UL approval for an AC power device. Their own device proper is an entirely low-voltage creature; and as such it breezed through UL listing since nothing lethal/dangerous needed to be tested for.  If you wondered why every darned electronics product in the whole wide world insists on using those annoying wall-warts, yup, that's why.
So simply buy a device like that, which comes with a wall-wart.  The wall-wart plugs in at an appropriate FIXED location. Then you either splice in the appropriate cabling, or maybe just wrap the supplied small cable in some wire loom (e.g. the spiral stuff) so it doesn't get chewed up by the drawer going in and out.  Run that cable to the moving drawer, and you're all set.