I'm sorry I can't recommend an ivy, other than to say that a small-leaved variety would look best. The heart-shaped backs will soon be overgrown, though (since they are not filled in, I am assuming that you want to trace the outline only) and you may find it a terrible amount of work to keep this looking the way you want. Consider a boxwood planted at the back of the seat. Take 2-6 leaders to trace the heart and maybe trim the rest to look like a throw pillow or bolster. You could fill in the entire seat with a moss, chamomile, thyme, or a trailing plant. Sounds like it might be cute.

I am thinking that the dark green leaves, deep pink very fragrant flowers
of Jasminum officinale would be a nice choice when you are out enjoying.
Plus the smell is heavenly.
Supannee

I was tickled to see your post. I did this with an old ice cream chair a couple of years ago. Removed the seat and used some chicken wire to make a 'bowl' in it's place. Lined the seat with moss, filled with potting mix and planted. I used a small leaf, varietaged ivy and added annuals for instant color. It's nestled under a family heritage maple and looks bright in an otherwise shady spot.

Here in Europe topiary tables, chairs and patio umbrellas are being comercially produced. The plant of choice seems to be Ligustrum.

Light annual vines might give your chair a boost - like Morning Glories? You could train one or two to climb around the wires.

What about planting some wild flowers behind and beneath them that will grow around them as opposed to on them then you can cut and shape your flowers to suit the effect your going for...a good one which has lots of foliage and plenty of pink and white flowers is campion rouge....or just scatter some wild flower seeds around the feature..hope this has been helpful...butterflies and bees love wild flowers....Happy gardening....