Most of the permanent/perennial vines (Ivy, Virgina Creeper, Wisteria, Trumpet vine, etc.) are going to be pretty "grabby" and/or too heavy for a drainage gutter to hold, so if you go that route, make sure the trellis is extremely strong & well supported. All of the above also have reputations for being quite aggressive - to the point of causing structural damage. A honeysuckle probably wouldn't be too bad, and you might see if there's a hardy jasmine that would work in your zone(?), clematis is also a bit better behaved, although it can be kind of picky about where it's planted.

OR you can go the annual route - some of these will also self-sow year after year so you won't have to re-plant them: Sweet Peas, Morning Glory, Cardinal Climber, Moon Flower, climbing nasturtiums, thunbergia alata, etc. The downside there is that they're dead and brown through the winter and require some clean up/removal of the dead stalks - hopefully spreading seeds at the same time.