Clay soil is usually nutrient rich, as opposed to sandy, light soil, and most climbers will be fine planted in it. Incorporate some humus rich material into the area before planting, using something like composted animal manure or garden compost or similar. The biggest consideration for climbers is how exposed the situation will be in terms of aspect and wind/sun/shade exposure and what part of the country you're in for the weather and winter temperatures particularly. Trachelospermum jasminoides, for instance, won't do well up North, facing North or East, with little sun and out in the open in a windy spot, but might do fine against a southfacing wall or fence in a sheltered spot.
You also haven't said how big your pergola is - some climbers get over 30 feet in height and 20 feet wide, others 8-12 feet high.

Roses usually do well in clay.

I have a rose 'Francois Juranville' growing in heavy clay over my little arbour (see photos). Most roses do very well in clay but do need some sun. I tip a couple of buckets of well-rotted manure around the root whenever I can get hold of it.