Is it possible to install the water line for a new refrigerator through cabinets to bathroom sink?

My refrigerator needs replacement and I really want one with a water dispenser, but want to make sure I can use it before I buy one (without too much trouble).

The fridge and kitchen sink are on different walls, so I've ruled that out. There is a half bath behind the wall behind the refrigerator. The bathroom sink sits on top of a cabinet in which there is cold and hot PEX lines each with a shut-off valve.

I've looked up some how-to videos and none of the setups look like mine, usually the piping is copper.

Is this something that can be done (maybe not by me, but by a professional)? Or would I have to go through the basement...which is completely finished and I don't know where the water lines are.

I'm newer to home ownership and looking to become more handy.