The SDSS weak lensing catalog used for many weak gravitational lensing studies is publicly available. Release notes describing the dataset and how it can be downloaded may be found here.
The GREAT3 challenge was a community challenge by the weak lensing community to test methods of inferring weak lensing shear from images. All of the challenge data are still available as a reference. Here are some key links associated with it:
From this page, you can access data from several of my papers. Any data that goes into figures in my papers is available upon request, but here are links to the ones that are requested most frequently:
To request an additional dataset, send me an e-mail or open an issue on GitHub. As a matter of policy, I will freely provide data points for all figures from all of my papers.