Does SCOTUS and the judicial branch have any way to actually enforce
laws on the executive branch?
Yes, although in practice, it is almost always trial courts and not the U.S. Supreme Court, that enforces orders.
The most common approach if an executive branch official defies a court order is to hold the official in contempt of court subject to incarceration and/or daily fines until the official complies or resigns. Typically, the lowest level official with the power to comply with the order is targeted for contempt of court charges.
This is not a vanishingly rare procedure and was done, for example, in this case in October of 2021: "The judge ordered Quincy Booth, the director of the city's Department of Corrections, and Wanda Patten, the warden of the DC Jail, to be held in contempt of court" (although neither incarceration nor a fine was imposed in that particular case). Also usually the government official can end to ongoing punishment by resigning from office and thereby depriving the official from the authority to carry out the order. This said, it very rarely reaches the point where a government official is personally punished in any meaningful way for disobeying the court order:
If any corporate entity disobeys a judicial order, the corporate
officer responsible for the entity’s disobedience can be sanctioned
for contempt. This would seem to imply that the federal official
responsible for an agency’s disobedience can be sanctioned, including
by imprisonment. The Justice Department itself appears to accept this
view. To be sure, it hasn’t always done so. Back in the 1950s, DOJ
contended that individual federal officials are absolutely immune to
contempt sanctions for official acts, similar to their absolute
immunity to common-law tort damages. But in more recent briefing in
1997 and 2008, DOJ distanced itself from this blanket immunity claim.
That said, there is another line of argument — less categorical and
more prudential — for the government to fall back on: that a court
should not sanction a person who makes all reasonable efforts to
comply. The concept of reasonable effort must be understood in light
of the nature of the noncompliant organization and each official’s
position within it. Recall that compliance problems are most common
when the judge’s order tells the agency to do something costly or
complex. That kind of order implicates the agency’s resource
limitations, its competing legal mandates and priorities, and the
question of what kind of technical and scientific information it needs
to acquire to formulate its action correctly. A judge assessing how
the agency handles these issues ends up having to make complicated
judgments about agency management. Identifying an official as
blameworthy for noncompliance entails an especially fraught judgment
of this variety: the judge must figure out which official(s) should’ve
acted, and what the official(s) should’ve done. To be sure,
higher-level agency officials — up to and including the agency head —
are formally responsible for whatever happens below them, but before a
judge could imprison an agency head or other manager for
noncompliance, she would probably need to find that actions promoting
compliance were reasonably within that manager’s grasp and that the
manager failed to take them.
In light of these prudential concerns, the judiciary has shown great
reluctance to imprison officials. Imprisonment has occurred only
twice, never for more than a few hours, and in both instances, the
biggest losers proved to be the imprisoning judges, one of whom was
thrown off the case for bias, while the other recused himself to avoid
a similar fate. Cases coming near to imprisonment are almost equally
rare. In the most spectacular of these — and the only federal agency
contempt case of any kind to get near the Supreme Court — the Commerce
Secretary in 1951 engaged in unusually clear disobedience when told to
return the shares in a bailed-out company to their private owners.
The D.C. Circuit ordered the Secretary jailed, only to have the
Supreme Court stay the sanction at the eleventh hour — and grant
certiorari in the case despite its previous refusal to do so. Justice
Robert Jackson wrote separately to say the Court should’ve demanded
immediate adversary briefing on the stay rather than simply grant it
indefinitely — a move that, in his view, rewarded the government for
its disobedience and signaled that the Justices had no stomach to use
force against official lawbreaking. Soon after, the plaintiffs
settled for a 50% discount, and the case basically disappeared from
memory, setting no precedent. Doctrinally, the door remains open to
imprisoning federal officials, yet judges’ prudential reluctance has
proven extremely strong.
This is generally carried out by the U.S. Marshal's office, which while located in the U.S. Justice Department, which overall is in the President's chain of command, but directly reports as a practical matter mostly to the orders of the judges of the federal district court to which the marshal is assigned.