You can extract the elemental values and perform the integration manually, but you will definitely need to design the mesh systematically so that they are traceable, i.e. in a simple square-ish or round-ish contours. Technically, you can define J from the contours of the moving crack tip.
Attached a reference, one of the most cited for J from DIC. You can use the elemental stress values from Abaqus instead of inferring them using compliance, which further simplifies the process.
Becker, T. H., et al. (2012). "An approach to calculate the J-integral by digital image correlation displacement field measurement." Fatigue & Fracture of Engineering Materials & Structures 35(10): 971-984.