A pergola climber is a type of VINE that is grown to climb up and over a pergola structure. It is usually a fast-growing, woody VINE that can be trained to grow in a particular direction. Common varieties of pergola climbers include clematis, wisteria, and honeysuckle. The term 'VINE' is an appropriate four-letter solution for this crossword clue as it accurately describes the type of plant used for this purpose.