It's possible, I suppose practical and presumably code. Kinda a weird setup, though. Kinda like using one electrical plug with a power strip because there aren't enough outlets.

I guess it would be more logical to put the coffee maker next to the sink, but that water supply is already supplying the hot water dispenser, and it's on the opposite wall from where I'd like to put the coffee maker. Moreover, the water lines to the refrigerator and coffee maker would be hidden if I use the washing machine water supply.

Would a copper tee connected to the existing 1/4" male threaded fitting work? The tee would supply both 1/4" lines to the icemaker and coffee maker.

I'd like to have a single cutoff for the ice maker & coffee maker, so I wouldn't have to disable the washing machine if a repair was necessary. However, because there are two hose bibs already on the cold water supply (I don't understand why the first bib is there), it could be a crowded arrangement if I add another cutoff.

You can find the parts you need at any local hardware store and they are relatively cheap.
Parts You need:
2ea. 1/4" compression needle line stop valve
1ea. 1/4" compreesion Tee
1/4" copper tube to make the additional lines.

Tools you will need:
Tubing cutter to cut the copper. (I prefer a mini cutter for 1/4")
Two wrenches that fit the valves..(two 8" adjustable wrenches would do fine)

Cut the pipe where you want the tee to the first appliance. Remove an and cap off the tee but be careful, there is a tiny brass ferrule under that cap that you need. First slip the cap on the copper with the open side of the cap pointing towards the cut end of the copper tubing. Now slip the ferrule on the tubing. (the ferrule looks like a miniature mans wedding ring).
Insert the end of the pipe into the fitting as far as it will go, then slide the cap forward, pushing the ferrule up against the end of the fitting and screw the cap on the end of the fitting finger tight. Now hold the fitting steady with one wrench while you tighten the cap with the second wrench.

Follow the diagram in the illustration I am attaching and use the same technique to tighten all the fitting caps.

I forgot a couple points above...

When tigthening the compression caps make sure you are holding the fitting body with one wrench while you turn the cap with the second wrench....Do not hold the cap on the opposite end of the fitting...

When you buy your fittings I would strongly suggest you buy 2 extra 1/4" ferrules. They will only cost you about a nickle each and they are very easy to drop or misplace. I would much rather have a spare part when I am done as opposed to running back to the store for a nickle part.

Wow -- you really need to change your name. I'm thinking getupandgopup or goingaboveandbeyondtohelptotalstrangerspup.

What an incredibly detailed tutorial you gave for my specific dilemma and with a diagram, too. Thank you for sharing your expertise and taking the time to explain it to me. I'm really so grateful. This forum is terrific. I mean it!