Again, legislated and also you are talking about the US Army. I can see that being the case. Shoot immigrants (all except the illegal ones apparently) have to go through a myriad of shots upon entry. I had to be vaxxed to go into the school system. But all that had been legislated in.
Now a court may up hold it and then there is nothing that one can do, but it will still has that hurdle (or they don't hear it and have the same net effect, until something later comes along that they do hear)). I am aware of one case, I think it was in 1905 and it was Mass. That was a state issue though, not federal.
Of course, there are also exceptions, although the exception of a natural immunity (from having previously contracted covid (now this is in Canada, but I do believe one person has won a case in that regard) or even an exception from those that their doctor is telling them not to get it due to health concerns (have a previous issue that would cause a higher risk for severe side effects etc, then there is usually the religous exemptions as well), I do not believe is in this mandate. Then, of course, what makes someone with a company of 99 people at less of a risk then someone with 100 people under their employ. It seems a little arbitrary other then it would fall under OSHA. And have a little more leg to stand on then if he went high on the hog and did everyone.
I find it strange, why wasn't the states involved in this? Why didn't he mandate states employees or withhold federal aid (now some states have mandated this and I do believe that there is a lawsuit in WA state right now over it)? Biden did threaten to withhold the vaccine if states didn't comply with what he wanted earlier, why didn't he do it with the vaccine now as far as going after states (or that may come later)?
Also, EO here has kids (depending on what the parents want to do) can go to school without masks. I know a few schools that are going against that. So while an EO is more binding then an EA, it's not quite as safe/binding so much compared to if it went through the legislative process.
Now, here is the thing with an EO, it's a little easier for the next person to come in and do away with it. There requires more of a push if it was legislated in.
I remember when a vaccine recently (as in recent memory) was mandated for school kids back in what 2007 TexasSignmaker, was that when he mandated it. That didn't turn out too well for him. By the way, that was an EO as well.