In theory, yes, you could use blackberry vines. BUT I have a rule of thumb: never put anything into the compost pile that you don't want more of ...
For myself, I'd be inclined to start small. My chief concern is those cane tips that touch the ground and sprout roots.
Everything depends on the size of the pile - if it's small enough for you to turn by hand, you'd be able to observe whether cane tips were rooting. If you have a reliable chipper/shredder or grinder, that would really grind them down into small sizes, that would be a big help.
What else are you composting? If you're adding shredded blackberry to a manure pile that gets turned, say, monthly, that would help them to break down. If it's kitchen waste from a family of 2-6, you'd likely want to add nitrogen.
I'm sorry I can't give a straightforward answer to your simple question. OSU Extension is hosting On Tuesday June 9 from 12 to 1. I'm the featured speaker for a call-in type composting Q and A.