It will only work if there is enough room behind the drawers to run the plumbing so they can cut through the cabinet to reach the DW. There's also usually an electrical outlet below the sink to handle the DW and Disposal, so make sure the cord from the DW is long enough or you'll need another electrical outlet..

Can the plumbing be run under the cabinet where the toe kick is? Drawers usually start 3-4" off the ground.

You can do it.

You just need water supply & discharge + electric to go behind or underneath the 15" cabinet. You are in the planing stage, so hardly an issue.

I forgot to mention...the cabinet and dishwasher are on a peninsula...so there is no wall behind.

Are you sure this is your only option? I recommend posting your layout here (if you haven't already) and let the people here help you with the layout. It may very well be your only option, but you would be surprised at what we can come up with for challenging kitchens!

Here is a link that might be useful: FAQ: How do I ask for Layout Help and what information should I include?

It can be done. GFI outlet in toe of 15, feeds come through kick, discharges depends on diswahser specks-but hose will fit behind draers. Read the DW install specs to plan it.

The manual for the dw I plan on getting says you need a 2.5" hole in the bottom 4" of the cabinet. It comes with it's own junction box to be mounted in another cabinet. It seems like I can run the plumbing in the bottom of the cabinet and the plug in cord that goes to the junction box.

The dishwasher discharge must run to the top of the cabinet, not in the base. I'm pretty sure that's code.