Could Biden force people and the military to take the vaccine against their will?

No this is America not a tinpot dictatorship.

He can't force anyone to get the vaccine. He can recommend it but not force anyone.

Dont make him force anybody.

That could never happen unless DC and Puerto Rico become states, Congress ends the filibuster and the Supreme Court is expanded. Oh, wait...yikes.

No but, anyone who refuses the vaccine or refuses to wear a mask or practice social distancing is a threat to the health and safety and well being of society and should be arrested.

The military, yes. Then again, that's ALWAYS been military policy. Civilians? No. You don't have to. Now, if nobody wants you as a customer because they're afraid you may kill them just by being there, .... But you don't "have to". No. 

The military are required to get lots of vaccines.  The president can't mandate it for civilians. 

the general answer is a qualified 'no'.  However, employers can require their contact with the public employees to become vaccinated -- in fact, OSHA workplace safety laws may require them to do so

He is counting on the vaccine to do his job.

Military yes, civilians no.