In a dissent in the case of Laird v. Tatum, Justice William Douglas argued tangentially that the creation of an air force is legal as per Article of 1, Section 8 of the Constitution:
The Army, Navy, and Air Force are comprehended in the constitutional term "armies." Article I, § 8, provides that Congress may "raise and support Armies," and "provide and maintain a Navy," and make "Rules for the Government and Regulation of the land and naval Forces."
In other words, Douglas argued against a literal interpretation of the Section - an interpretation that I don't think has been disputed.
Going into more detail, this blog post talks about John Marshall's opinion in McCullough v. Maryland, where he gave conflicting statements about the Enumerated Powers listed in Section 8. The Enumerated Powers are, in a nutshell, those granted to Congress via Section 8. While several common passages of Marshall's decisions are often cited for different interpretations of his points, one in particular caught my eye:
Among the enumerated powers, we do not find that of establishing a bank or creating a corporation. But there is no phrase in the instrument which, like the Articles of Confederation, excludes incidental or implied powers and which requires that everything granted shall be expressly and minutely described.
Marshall also referenced the Necessary and Proper Clause in Section 8, which states that Congress can make laws necessary for the execution of the powers explicitly listed - which makes sense, backing up the argument of Justice Douglas. In particular, the clause states that Congress has the power
To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.
The application of this clause is strengthened, as lly pointed out, if we consider that the United States Air Force was previously part of the US Army, and therefore, the use of aircraft in war was already considered an extension of the army - and thus a necessity to raise an army. Therefore, it wasn't unnatural for it to be extended into a separate branch.
As an addendum, it seems to be accepted that Congress could authorize the creation of a "space force", according to domestic law. Arguments against the legality of such a measure don't typically invoke Section 8 - but that's a rabbit hole I'd rather not go down. The point is, a literalist interpretation of the Constitution is not being applied at present in the case of a space force, and likely would not be applied to challenge the existence of the United States Air Force.
In summary, in Laird v. Tatum, Justice Douglas argued that a broader interpretation of "armies" should be used, and while it was given in a dissent and therefore isn't precedent, it seems to be backed up by Marshall's broader interpretation of Section 8 in general, along with the Necessary and Proper Clause. As far as I know, though, the constitutionality of the Air Force has never been seriously challenged in a US Court.