Answer: It is too late to plant annual vines such as morning glories or moon flower vine, or cardinal climber or purple hyacinth beans, but these might be possibilities for next year.
Perennial vines would be better planted in September when the hottest part of the summer is over. In a hot and sunny location you could consider trumpet creeper (Campsis radicans) or wisteria, both of which are very large, vigorous, strong vines. In a spot with morning only sun you could consider clematis which is a smaller, lighter weight plant.
Keep in mind, in deciding what to plant, that you may need to access the lattice to maintain it with sealer or paint, and that lattice tends to work like a sail in windy weather and will be very heavy if covered with a vine. Large perennial vines can thicken and break apart the lattice over the years, so you may want to use annuals for that reason. Another option is to plant tall perenials or flowering shrubs along the lattice rather than grow vines on it. I hope this gives you some ideas.
Q&A Library Searching Tips