what is a cheap printer for just printing black and white documents with refillable ink cartridges?

there are no refillable ink printers that do black and white.

if you plan on printing tons of pages, get a laserjet printer.

otherwise, just buy a cheap inkjet. be sure you set it to greyscale so that it prints only black ink.

A bottom of the barrel model of laser printer from Brother, preferably one that's no longer in production and is on super low sale. You know you found one when the price of the brand new printer is lower than the price of the replacement toner cartridge.

a low end laser printer fits that bill and the toner costs less per page than refilling ink.  look for the $50 [US] model

Consider alternatives. 

Laser printers may be initially more expensive. But, if you do a lot of printing, toner will cost less than inkjet ink. You'll save in the long run.

You could also consider printers that use ribbons instead of ink. Impact printers can print on multipart forms. Daisy wheel printers have good quality printing, but you are limited to the characters on the wheel. Dot matrix printers with wire print heads are more flexible.

You can buy small thermal printers specifically made to print labels.

No such thing. "Cheap" printers are subsidized by the sales of expensive, proprietary cartridges. If you really need something cheap, get a thermal printer.