Overview
An executive order cannot make new law. However, most executive orders are based on powers granted by law to the President, or to some executive agency or department. Others are based on laws that come under the general power and duty of the President to "take care that the laws are executed" and announce some policy for how laws will be interpreted and enforced.
Announced mandates
In the case of the mask and vaccine mandates announced but not yet issued by the Biden administration, they apparently claim to exercise powers granted under various laws, particularly the Occupational Health and Safety Act.
It is likely that once such regulations are formally issued they will be challenged by those opposed to such mandates. If they are upheld, (or are somehow not challenged and thus assumed to be valid) they will carry the force of the laws under which they are issued, and thus the Supremacy Clause will apply to those laws, and to the orders as ways to enforce those laws.
Other Precedents
The question says:
Currently, the only precedent to enforce fines and vaccine mandates (at a state level) that I am aware of is a 1905 decision by the Supreme Court, Jacobson v. Massachusetts (197 US 11 (1905)), where it allowed Massachusetts to fine an individual for refusing to comply with vaccine mandates set by the state.
This is not quite correct. Jacobson is the leading case on this issue, but there have been some others. In Zucht v. King, 260 U.S. 174 (1922) the US Supreme Court held that a local ordinance mandating vaccinations for school attendance did not violate federal constitutional rights, citing Jacobson and calling the matter settled law. Note that the ordinance in Zucht applied to both public and private schools without exception.
In Prince v. Massachusetts, 321 U.S. 158 (1944) the Supreme Court opinion included (at 166) the statement that:
Acting to guard the general interest in youth's well being, the state, as parens patriae, may restrict the parent's control by requiring school attendance, regulating or prohibiting the child's labor and in many other ways. Its authority is not nullified merely because the parent grounds his claim to control the child's course of conduct on religion or conscience. Thus, he cannot claim freedom from compulsory vaccination for the child more than for himself on religious grounds. The right to practice religion freely does not include liberty to expose the community or the child to communicable disease or the latter to ill health or death. People v. Pierson, 176 N.Y. 201, 68 N.E. 243. The catalogue need not be lengthened. It is sufficient to show what indeed appellant hardly disputes, that the state has a wide range of power for limiting parental freedom and authority in things affecting the child's welfare, and that this includes, to some extent, matters of conscience and religious conviction. (Emphasis added, footnotes omitted)
Prince v. Massachusetts was a child labor case, not a vaccination case, and the above comment was technically obiter dictum (not binding precedent). But because of it Prince has several times been cited in later vaccination cases alongside Jacobson and to show that Jacobson is still good law.
Note that Prince, like Jacobson and Zucht, was a case supporting state law against a 14th amendment challenge.
Supremacy Clause
An Executive Order that is not backed by any valid law would not be the "Supreme law of the land" under the supremacy clause, and might well be simply held invalid for lack of Presidential authority to issue it, depending on the subject of the order. But orders claiming to make law on the President's own authority are quite rare.
Youngstown Sheet & Tube Co. v. Sawyer
Youngstown Sheet & Tube Co. v. Sawyer, 343 U.S. 579 (1952) (the steel seizure case) invalidated such an order. In that case the President not only did not have the backing of a specific law, but failed to follow the procedures set out in a relevant law.
In Youngstown Sheet & Tube Co. v. Sawyer the concurring opinion by Justice Jackson has proved influential in later cases and in later congressional drafting of laws. The key passage of that opinion starting at 343 U. S. 635 reads:
Presidential powers are not fixed but fluctuate depending upon their disjunction or conjunction with those of Congress. We may well begin by a somewhat over-simplified grouping of practical situations in which a President may doubt, or others may challenge, his powers, and by distinguishing roughly the legal consequences of this factor of relativity.
When the President acts pursuant to an express or implied authorization of Congress, his authority is at its maximum, for it includes all that he possesses in his own right plus all that Congress can delegate. In these circumstances, and in these only, may he be said (for what it may be worth) to personify the federal sovereignty. If his act is held unconstitutional under these circumstances, it usually means that the Federal Government, as an undivided whole, lacks power. A seizure executed by the President pursuant to an Act of Congress would be supported by the strongest of presumptions and the widest latitude of judicial interpretation, and the burden of persuasion would rest heavily upon any who might attack it.
When the President acts in absence of either a congressional grant or denial of authority, he can only rely upon his own independent powers, but there is a zone of twilight in which he and Congress may have concurrent authority, or in which its distribution is uncertain. Therefore, congressional inertia, indifference or quiescence may sometimes, at least, as a practical matter, enable, if not invite, measures on independent presidential responsibility. In this area, any actual test of power is likely to depend on the imperatives of events and contemporary imponderables, rather than on abstract theories of law.
When the President takes measures incompatible with the expressed or implied will of Congress, his power is at its lowest ebb, for then he can rely only upon his own constitutional powers minus any constitutional powers of Congress over the matter. Courts can sustain exclusive presidential control in such a case only by disabling the Congress from acting upon the subject. Presidential claim to a power at once so conclusive and preclusive must be scrutinized with caution, for what is at stake is the equilibrium established by our constitutional system.
Into which of these classifications does this executive seizure of the steel industry fit? It is eliminated from the first by admission, for it is conceded that no congressional authorization exists for this seizure. That takes away also the support of the many precedents and declarations which were made in relation, and must be confined, to this category.
Can it then be defended under flexible tests available to the second category? It seems clearly eliminated from that class, because Congress has not left seizure of private property an open field, but has covered it by three statutory policies inconsistent with this seizure ...
Would a vaccine mandate or mask mandate that goes beyond any law passed by Congress fall into Jackson's "zone of twilight"? In the absence of a court ruling, no one can say.
