Just so you know......most groundcovers are not all that great at weed suppression. And can become quite difficult to weed out successfully should the weeds get a toe hold.

Rather than a traditional "groundcover", I'd suggest you look at some low growing spreading perennials that can cover the ground yet still compliment your other plantings. Things to consider are cranesbills (hardy geraniums), dianthus (cottage pinks), creeping phlox, candy tuft (Iberis sempervirens), thrift (Armeria maritima) or various sedums/stonecrops.

Note: these suggestion are for full sun since your current plantings would indicate that's what this area receives.

Thank you! I will look into those.

What about Veronica?

I do also need it to be rabbit resistant.

I like Veronica prostrata. It's evergreen, and it forms a dense mat that deters weeds. It also produces beautiful blue-and-white flowers each spring. It's reportedly rabbit resistant, but I have a lot of rabbits where I live, and they have chewed on it some. Fortunately, it grows fast enough that it may be able to stay ahead of the rabbit munching. Another possibility is Veronica pectinata. Groundcover beautify garden

This area is in a corner of our yard where we share a fence with our neighbors. Will I need to worry about it going under the fence to his yard? Or is it easy to maintain?

It's easy to control.