Certified GKH Gardening Expert
Yes, you can transplant perennial wild sweet pea. Sometimes a very old plant won't make it, but it's worth a try. This plant can be transplanted in late summer.
Before transplanting, you can cut back most of the foliage but leave about 12 inches worth, to ensure the roots will be able to support all the foliage that is left. See this article for tips on avoiding transplant shock: