Well, there is Gelsemium sempervirens, Carolina jessamine. Blooms now so it attracts early pollinators, is evergreen, can be a bit aggressive.

Blooming late in the year is Clematis virginiana, Virgin's bower.

In between you have one of the native roses, Rosa setigera, it does have thorns but they are small.

Another choice is Decumaria barbara, sometimes called climbing hydrangea.

Here is a link that might be useful: Brushwood Nursery - good source for native vines

i used to grow many mini ;-) roses, one of which was 'jean lajoie' and it was very nice. it is pink flowered. there is a climbing version of it as well.
another one i have read about that would fit your color scheme better is climbing miniature rose 'work of art'.
the flowers open salmon orange with yellow reverse and they fade to a lighter orange. would look nice with your yellow honeysuckle. it is said to be thornless and repeats well.
from my experience, the mini roses did not have the disease problems typical of hybrid teas and bloomed intermittently all summer.

Here is a link that might be useful: 'work of art' mini rose

Ha! Jeff you are funny - I had work of art (and will try it again) but my neighbor accidentally weed sprayed it, and I also had Jean Lajoie for quite some time. I really liked her but she was a bit of a black spot magnet for me. I just couldn't find a way to control it.
Native clematises are possibilities. Since this is for a vegetable garden I do need something that will not spread by seed...
I'm trying that rose from seed, Esh. No luck with germination so far and my other roses which tend to seed out have already germinate and are small plants...so I don't know what is going to happen!!!

How about Aster carolinianus, climbing aster? Sounds ideal, but have no personal experience of it.

GGG, have you chosen a vine yet? When I read this, potato vine (Solanum jasminoides) popped into my mind--if you're warm enough there. I really miss it from SoCal. You probably know it, but it's a tough pretty plant that shows up on arbors everywhere, carrying the show on even if the stars are throwing a tantrum. Not too large, easy to keep to size, light dainty foliage and sprays of small white/yellow centered flowers month after month (most of the year in SoCal, anyway). The flowers like to stick to you if you brush against them, but the only real thing keeping this little thing from perfection in its way is that the flowers aren't fragrant.

Hi again!! We had no internet after the storm until today, so sorry for slow answers....yes, I do have a variegated potato vine that HATES it's location, Rosie, so it is a very very good idea to move it and to let it do it's thing on there. It will be perfect. Good idea!!
Yes scmatlanta, I do have a carolina aster. and it's not really a vine. It is indeed an odd bird, and likes to sort of "throw" itself over a fence, sort of like a dramatic leading lady in the middle of a heartbreaking scene throws herself over someone's shoulder to cry. That is the habit of the Carolina aster... :)

Variegated! That sounds very nice. I'd forgotten, but last year our Carolina aster collapsed and wept on me when I tried to take a fence down. I kind of tossed her onto the handy but sadly indifferent bosom of a nearby eleagnus. If that shrub doesn't smother her entirely, the flowers should be pretty against the silvery foliage.