Water – the missing part that would fix this outdoor spigot

First confirm that the problem is at the valves and not a clogged faucet screen.

Backing-up the valve with an adjustable wrench you could turn your knobs with channel locks but be prepared for a few drips, or something completely catastrophic to happen. Not something to be messing with on a Saturday evening. Being frank, those valves are crap. Preferably use 1/4 turn valves. Seeing that these are compression fitting, they're easily changed. Ask for these.

The choke point is at the valve, this clears the blockage: Once you have the valves spinning freely, shut them off and disconnect the leads at the faucet (back this up carefully, do not twist the copper leads). Run the lines into a bucket, completely opening and closing the valves several times. Reattach them, turn them on. Do not touch them again without repeating these steps. Remove the faucet screen and run some water for a bit. Reinsert a cleaned screen, done.

The blue plastic piece itself can't be replaced. The whole inlet valve needs to be replaced. Pretty sure it's this part which costs about $50. Not very familiar with taking apart dishwashers down that far but the appliances I have taken apart have been surprisingly easy. I've always found videos online that helped. This video looks like your dishwasher. https://www.youtube.com/watch?v=MNMd_lDdG_w looks very easy... but then again so does screwing in a supply hose. :)

Considering it's a new appliance and your inexperience with repairing dishwashers you may want to call Whirlpool. It may even be covered under warranty.

Very surprised to see that it's made of plastic.