It seems the most specific analysis with respect to EUAs (emergency authorizations) is provided a long Lawfare article.
Basically, unlike traditional FDA approvals (made under a 1938 law), EUAs are made under a newer (2004) Project Bioshield Act. The Act authorizes the HHS rather than the FDA to make these EUAs, but the HHS has in turn delegated this authority to the FDA, while retaining a sort of emergency override.
However, the Project Bioshield Act still provides for fairly restrictive approval criteria that are tied to scientific questions:
the secretary may then issue an EUA “only if” the secretary concludes that “based on the totality of the scientific evidence available” about the product, “it is reasonable to believe” that it “may be effective in diagnosing, treating, or preventing” the given “serious or life-threatening disease or condition,” and that when used for this purpose, the “known and potential benefits of the product ... outweigh [its] known and potential risks.” The secretary must also find there is no other “adequate, approved, and available” alternative to the product.
While there has been at least one case in which an FDA/HHS rejection under the 1938 law has been successfully challenged in court, that doesn't seem to be the case for the 2004 Bioshield Act. In fact, a challenge in court of the EUA for hydroxychloroquine failed; somewhat confusingly it was the withdrawal of the EUA that was challenged. On the other hand, the article notes that that challenge was poorly framed (under the Administrative Procedure Act, which explicitly does not apply to Project Bioshield) and that based on the D.C. Circuit's decision (in another 2020 case, not related to drugs) an ultra vires challenge to such EUAs (e.g. if completely unscientific) is still possible, even outside the Administrative Procedure Act.
So, it may be harder to challenge EUAs in court than regular FDA approvals, but not impossible.
Because it falls under the HHS, the FDA is not an independent agency and there have been a number of calls/proposals over the years to make it more so:
In 1994, the Social Security Administration was made independent from HHS to skirt political pressure. Today, amid a national health crisis that relies on both patience and scientific rigor to fight back, the FDA should be granted the same autonomy. The idea is not new; in 1975, the Senate Committee on Labor and Public Welfare called for granting FDA independence, and in 1991, a Blue Ribbon advisory committee on the FDA recommended redelegation of regulatory autonomy and establishment of the FDA as a freestanding agency.
The article goes on to discuss in more detail a more recent (2019) proposal, which would involve commisionners like e.g. the FTC has.
As a reminder, in US law/jurisprudence (since 1935) an "independent agency" (such as the FTC) means that:
Such a body cannot in any proper sense be characterized as an arm or an eye of the executive. Its duties are performed without executive leave, and, in the contemplation of the statute, must be free from executive control. To the extent that it exercises any executive function -- as distinguished from executive power in the constitutional sense -- it does so in the discharge and effectuation of its quasi-legislative or quasi-judicial powers, or as an agency of the legislative or judicial departments of the Government.
Of some note however, under a 2020 Supreme Court decision, the POTUS can now fire some "independent" agency chiefs (such as the one of the Consumer Financial Protection Bureau) making their status a bit more blurry in US law.
In order to ensure the CFPB's independence, the law creating the agency called for it to be headed by a single director, confirmed by the Senate, who would serve a five-year term and who could only be fired for malfeasance, inefficiency or neglect of duty.
The decision was a victory for President Trump and for forces in the business community that have long sought to trim the sails of independent regulatory agencies, from the CFPB to multimember-led agencies, among them the Securities and Exchange Commission, the Federal Reserve Board, the Federal Communications Commission and many more.
Writing for the majority, Chief Justice John Roberts said the "the CFPB's leadership by a single individual removable only for inefficiency, neglect, or malfeasance violates the separation of powers."
In its Seila  decision, the court’s majority said the exemption in Humphrey’s Executor  that allows the president to fire the leaders of an independent agency only for cause still applies to “multimember expert agencies that do not wield substantial executive power,” and to “inferior officers with limited duties and no policymaking or administrative authority.”
According to various commentators, further litigation in this area is to be expected, because of the various interpretable phrases in the latest US Supreme Court decision on this matter, e.g. what is "substantial executive power". (Even if the FDA were an independent agency by statue [which it isn't], would drug approval count as "substantial executive power", under the 2020 Supreme Court ruling, for instance? If so, in a hypothetically reformed FDA as an "independent" agency, the president could [still] fire all FDA commissioners for any reason[s].)
Actually there have been concrete efforts by the White House to change various guidelines etc. issued by both the FDA and (even more so) by the CDC (which also falls under the HHS) this year.
However, the 1938 FFDCA law that sets the criteria for drug approval decisions has apparently been used successfully to challenge at least one FDA/HHS decision in (federal) courts (Tummino v Hamburg 2013).
So my guess is that if the FDA/HHS were to approve drug or vaccine way outside the usual process, it could be challenged in courts in view of that 2013 precedent, as a first step.
(I think such limits on delegated powers, e.g. requirement that approval follow some scientific criteria are constitutional--@phoog commented to that effect as I reading up on this. In fact, such limits/guidance actually seem[s] necessary in order for a law that delegates powers to even be constitutional.)
As for the (leading) Covid vaccines, these probably would not fail a judicial review of the evidence if e.g. some corner were cut in the approval process at this (late) stage (seeing that one was e.g. approved in the UK and that the US approval process is probably the most elaborate/demanding worldwide), but I'm guessing the political hooha that could come from litigating in court something like that might not be worth the hassle (especially to this administration, and now, after they lost the election).
Note that FDA's own rules for such (emergency) approvals are still pretty demanding (despite the word emergency):
Under the FDA’s rules for COVID-19 vaccines, companies can apply for an EUA when half the trial participants (half of 43,000 people in Pfizer’s case) have been followed for two months after their last dose.
If these criteria were drastically changed and a FFDCA lawsuit were brought challenging the new criteria, presumably a court would look at criteria for prior (emergency) FDA vaccine approvals in weighting
whether a change in criteria was "arbitrary, capricious, or unreasonable", or otherwise it would have to agree with some expert testimony that the change is justified by the pandemic circumstances etc.
As further background, it seems the safety & efficacy data for non-emergency FDA vaccine approvals is at least 6 months and can well span 1-3 years.
(Interstingly enough, the FDA is being sued over its GRAS rules, claimed to be unscientific, so a violation of FFDCA.)