Congress can pass legislation to end or defund the programs
In the United States, Congress determines what the military is allowed to do and provides funding for those activities by passing legislation every year. Specifically, the National Defense Authorization Act (https://en.m.wikipedia.org/wiki/National_Defense_Authorization_Act) appropriates money for the Department of Defense. The Department of Defense (or for that matter, any part of the Federal Government) cannot spend money on any activity that Congress has not appropriated funding for. Given that the diversity programs are carried out by people paid to give instruction, and are attended by people who are paid as federal employees to go to them, a removal of funding for these activities would effectively ban them from taking place within the military.
It would be a trivial legislative exercise to amend the next NDAA to remove all authorization for funding any of these diversity programs, provided there are sufficient votes among Congressmen to do so.
What is "jurdicially legal" can be changed by Congress through the passing of legislation
Senator Cotton and Representative Crenshaw are proposing to do just that.
Crenshaw and Cotton can do anything they want with the "reports"
They're probably just going to collect a bunch of anecdotes to support the position they've already decided to take against these programs they don't like. It doesn't really matter.