Is the structure a solid fence (no gaps, or very small gaps, between boards) or open? Different species of vines have different means of climbing, and some can cling to solid surfaces while others need something to wrap around. A solid fence could accommodate the latter group if you anchor guy wires or some other slender support to its surface, held about a couple inches away to give the stems room to twine.

Most vines are aggressive growers and produce prolific growth, so in a sense, most will create a "mat" of foliage since it's in their nature to cover a surface since that's their only way of getting enough light exposure.

We do not have veterinary training and cannot speak to plants safe for dogs; you can check with a vet or perhaps an ASPCA toxic plants list to verify if any species we suggest are risky or fine.

Clematis is a diverse group, and only a few "take over" and mature quite large. We would not recommend planting Sweet Autumn Clematis (Clematis paniculata / terniflora) since it's invasive, but most others on the market are fine to grow and only get about 8 to 12 feet in length/height. However, Clematis need a slender support to wrap their leaf stems around, and cannot climb a solid wall on their own without needing to be tied-up to something as they grow.

Many vines thrive in full sun, as long as the soil drains well. You don't mention how much space is available (vines often get as large as their support structure, however tall or wide it is). If you need the plant to be deer-resistant, that will limit your options, but here are some ideas to get started:

• Virgin's-Bower Clematis (Clematis virginiana) -- native; white flowers in late summer
• any number of climbing varieties of Rose (there are dozens of varieties on the market); a small handful, like Rosa banksiae and cultivar 'Zephirine Drouhin', are essentially thornless
• Virginia Creeper (Parthenocissus quinquefolia) -- native; it blooms, but flowers are not terribly showy, and it's grown more for the autumn foliage color instead; birds enjoy the berries; will cling to surfaces on its own; matures large, but can easily be cut back as needed
• Trumpet Honeysuckle (Lonicera sempervirens) -- native and, unlike the many invasive honeysuckles in our area, not scented; popular with hummingbirds

A list of invasive species to avoid planting is included on our Vines for Maryland Gardens page. Note that most vine species on that page mature larger and have denser growth than it sounds like you want. While some are easy to trim smaller, doing so would likely reduce their flowering display.

Miri

Thanks for your recommendations. Will definitely consider adding supports to secure vine to surface. 
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You're welcome.