Who, if anyone, is trying to stop the mandate being foisted on our military?
There are multiple lawsuits on the military mandate, and on the EO (contractor) mandate. I don’t have a list, but a websearch should find a list of cases.
I understand Pacific Justice Institute has an attorney dedicated to the military mandate.
https://childrenshealthdefense.org/defender/lawsuits-dod-covid-vaccine-mandates-us-military/
7 Lawsuits Challenging COVID Vaccine Mandates for U.S. Military Members