This is the site I use to check plants for toxicity.

https://www.aspca.org/pet-care/animal-poison-control/toxic-and-non-toxic-plants

ETA: I have found it quicker to just to google using the name of the plant and dogs.

Sun or shade or a mix? Planted on the dog side or the other side? Kind of fence? Vines cling different ways, so some may work better on certain kinds of fences.

Here is a similar question from the Ladybird Johnson Wildflower Center that gives both suggestions and resources where you can look up plant toxicity for dogs.

https://www.wildflower.org/expert/show.php?id=10819

Of their suggestions, I have grown Lonicera sempervirens, a native long-flowering honeysuckle with red, orange or yellow flowers. The cultivar I have, ‘Major Wheeler’ blooms heavily in May (mid spring for me, but it would be earlier for you) and then less heavily all summer. It would be semi-evergreen for you. For me a lovely and easy care plant. However, it climbs by twining so couldn’t climb a flat fence without wires to wrap around. I also have Virginia creeper, Parthenocissis quinquefolia, which is deciduous and has unobtrusive, non-ornamental flowers. It can grow up either a wire fence like chain link or a board fence. Either may need some guidance to grow to fully cover the fence.

You might also consider a row of bushes along the fence as an alternative to vines.

https://www.hunker.com/13428041/what-trellis-plant-is-not-poisonous-to-dogs

I am currently changing out all my houseplants, because my old cat decided to eat them. I have also got to redo the back yard, because our new dog is sampling everything.

I think I will try Major Wheeler honeysuckle. It seems like it fits the entire bill for what I am looking for, plus it can be one of the new things I try this year (I like to try something new each spring).

I did do my research and I am now looking for a supplier. Does anyone out there have a goods recommendation?

Thanks again!

Woops I just discovered Major Wheeler honeysuckle is toxic to dogs..........

I'm finding lots of conflicting information on this subject

The Coral Honeysuckle Lonicera sempervirens (Coral honeysuckle) is a handsome evergreen vine that is a fine addition to your landscape. When we are asked about toxic plants, we usually go to the databases listed below. We did not find Lonicera sempervirens in any of the lists which is a good indication that it is not toxic. The Canadian Poisonous Plants Information Sytem listed two other species of Loniceraas being toxic: Lonicera japonica (Japanese honeysuckle) and Lonicera tartarica. In addition, both of these are non-natives and are considered invasive species in the U.S.

Toxic Plants of Texas

Poisonous Plants of North Carolina

Poisonous Plants of the Southern United States

Cornell University Plants Poisonous to Livestock

University of Pennsylvania Poisonous Plants

Canadian Poisonous Plants Information System

California Poison Action Line

Florida Poison Information Center

ASPCA

Aggie Horticulture

Honeysuckle does not appear to affect birds, wild animals, goats and most horses. The California Poison Control System does not flag honeysuckle as a poison, and specifies that winter honeysuckle (Lonicera fragrantissima), which grows in USDA zones 4a to 8b, is not toxic. Other authorities that do not count honeysuckle as poisonous include the American Society for Prevention of Cruelty to Animals, the University of California Davis School of Veterinary Medicine, the University of Illinois Veterinary Medicine Library, and Cat Fanciers. However, other organizations, such as Pet Education and Dog World, warn that the toxins in honeysuckle vines and berries retain the potential to harm pets. The Cornell University Department of Animal Science and the Wildflower Center warn that the berries of Japanese honeysuckle and the European varieties Graham Thomas (Lonicera periclymenum) and tatarian honeysuckle (Lonicera tatarica), found in USDA zones 4a to 8b, may also harm humans.

It appears that the European and Japanese ones seem to be the ones causing the problems.

The Major Wheeler is a variety of Coral Honeysuckle. Be sure to check the botanical names, as the common names vary.

Coral honeysuckle is a selection of Lonicera sempervirens. IME it produces little in the way of fruit, though it does sometimes have some.

I stated the botanical name in the post with the links. I didn't see the need to restate it in the next two.

The one you suggested that the OP liked, "Major Wheeler" is just a specific variety of Coral Honeysuckle. one or two sites are reporting it poisonous to dogs, but I believe that they are just lumping all honeysuckle in one item, no matter that there are several different species.

Thanks again everyone for all of your advice!

I am trying to figure out the best/easiest way to deal with fence that has to stay where it is and also not be a problem for one of my young dogs who thinks everything she sees, smells and hears is appropriate for her mouth.

Information on the web is confusing concerning what may or not be toxic to dogs. Given my pups eating habits I am inclined to not plant anything that may be questionable.

Thanks again!

Ive been on the hunt as well. The only one I've found that is non-toxic is Hardy Kiwi vine. You need 2, a female and a pollinator(male). What did you end up planting?