If your are talking Himalayan blackberry, bushes that grow higher than head high, those chip up great. Western pacific blackberry, which only grow ankle high, and are not usually considered a problem, can be a bit more of a problem.
What I typically do is start chipping woody material, and save that aside in a wheelbarrow. Then when running the shredder, if it starts to gum up (you will develop a feel for it) you can then add a shovelful of woodchips, to dry up and unclog the shredder.
Careful though, when putting blackberry vines un the shredder, sometimes it can self-feed at speed, whipping sharp thorns past you. For that reason, its best to deal with blackberry in the winter, when its easiest to be bundled up against the thorns.