"... from the back of the house (from the gutters above, plus a pipe under the small bit of the house from the side yard to the back yard." That's clear as mud. :-) If the two drives shown on the plan are the ones in question, it looks like they'd be draining only the alcove/courtyard area at their head ... not water from the back of the house. If something unusual is going on, you need to explain clearly and probably show pictures that corroborate the explanation.

To be sure, you'd need to set back a raised bed some number of feet (probably 4' at least) from the edge of each drive, so that there is adequate room at each side for door opening and pedestrian circulation.

You're not giving any clues about how this will look, but I have a hunch, it's not good. Maybe I'm wrong. Is the neighbor on board with the plan?

In order to not stoop while gardening, a bed would have to be quite high. That's going to take a lot of soil. If ever a day comes where you, or the neighbor, is completely sick of the raised bed, it's not going to be fun to undo and clean up. Are tomatoes and squash worth it when you can buy them at the store for a infinitesimal fraction of what this project is going to cost? I have a bad back and don't like to stoop for gardening or anything. But I still garden on plain old ground. I kneel sometimes (on an old kitchen rug.) And I sit. And sometimes I stand. There's not any place or time I ever considered bringing in truckloads of earth in order to bring the ground up to me. Consider if doing this is a good idea.

Thanks for responding. I'll try to get a better picture up today.

Both of those driveways are mine; no neighbors are implicated in this. The one on the left is our old driveway to a garage that is now converted to a dojo. The one on the right goes into the current garage. We actually use the garage for cars, so it is very rare that we get in and out of cars in the driveway, and if we do there is only one car at a time.

And yes, I understand how hight the bed would be. We do have the dirt for it already. There is no reason to bring in more. This is the specific goal of a keyhole garden.

As for whether the tomatoes and others are worth it. I plan on using it for heirloom vegetables, not store bought stuff. I live in a very diverse area and I still cannot find heirloom vegetables.

Lastly, I have some arthritis in multiple joints already and I will probably decline as the years progress.