Is there such a thing as a rechargeable keyboard?

I have never seen a keyboard that required a battery at all, let alone a rechargeable one. They either are wireless--run on a Wi-Fi signal--or they have a usb cord. Or both. 

yes there is , its called a wireless keyboard . plenty of brands too , check out the computer shop .

WIRED MOUSE AND WIRED KEYBOARD NOT NEED TO BE RECHARGED:

RECHARGEABLE DESIGN: Both of the keyboard and mouse use built-in lithium battery, can be easily recharged through included USB cable. And the battery is incredibly long-lasting — it will power your keyboard and mouse for about a month or more between charges.

Amazon has hundreds from full sized to ones similar in side to a video game controller.  Make sure you get one that works with your computer.  Many of them are Bluetooth and if your computer doesn't have Bluetooth it wont work.  The keyboards with the USB receiver will work on pretty much any computer or newer Android device with an extra OTG adapter.  

YES Logitech and Microsoft both have rechargeable wireless keyboards, I prefer ones that light up the keys...

Yes, I have a Cherry DW9000 slimline keyboard plus mouse set, both rechargeable by USB.

And they are switchable between bluetooth, or using a USB receiver stored in the mouse.

Keyboard for my tablet is rechargeable, nothing new there.

Try searching, that would help.