At this time, this is not something we currently offer at Gradescope, but we do have this feature request listed on our roadmap. You can vote and add comments on this feature request here: Lockdown browser functionality on Gradescope. We'll then follow up with you if and when this feature becomes available on Gradescope in the future.
If you are looking to support your high security exams with device control tools, Turnitin has other solutions that might be a good fit, and so would advise raising this with your Account Manager.