From my understanding of the law, the incumbent VP is exempt and as recently as 2000 has campaigned, but other federal employees cannot "be candidates in partisan elections". That statement, which I got from Wikipedia and which does not appear in the actual bill text, seems pretty iron-clad, but then there have been modernization acts and such since 1939. Also, if I'm not mistaken, presidential administrations are free to tighten or loosen Hatch enforcement. All things considered, would a sitting cabinet secretary face the sort of fundamental issues that keep a campaign from getting off the ground, for instance denial of ballot access in the states?
No, they are not. Under the Hatch Act, 5 U.S. Code § 7323,
(a) Subject to the provisions of subsection (b), an employee may take an active part in political management or in political campaigns, except an employee may not—
(3) run for the nomination or as a candidate for election to a partisan political office;
Subsection (b) imposes additional restrictions for employees of certain agencies (like the CIA, FBI, or FEC). As for definitions, § 7322 says that:
(1) “employee” means any individual, other than the President and the Vice President, employed or holding office in—
(A) an Executive agency other than the Government Accountability Office; or
(B) a position within the competitive service which is not in an Executive agency;
but does not include a member of the uniformed services or an individual employed or holding office in the government of the District of Columbia;
(2) “partisan political office” means any office for which any candidate is nominated or elected as representing a party any of whose candidates for Presidential elector received votes in the last preceding election at which Presidential electors were selected, but shall exclude any office or position within a political party or affiliated organization;
There is one exception, found in § 7325:
The Office of Personnel Management may prescribe regulations permitting employees, without regard to the prohibitions in paragraphs (2) and (3) of section 7323(a) and paragraph (2) of section 7323(b) of this title, to take an active part in political management and political campaigns involving the municipality or other political subdivision in which they reside, to the extent the Office considers it to be in their domestic interest, when—
(1) the municipality or political subdivision is—
(A) the District of Columbia;
(B) in Maryland or Virginia and in the immediate vicinity of the District of Columbia; or
(C) a municipality in which the majority of voters are employed by the Government of the United States; and
(2) the Office determines that because of special or unusual circumstances which exist in the municipality or political subdivision it is in the domestic interest of the employees and individuals to permit that political participation.
These restrictions apply both on and off duty. The objective wasn't just to say "federal employees can't use federal authority for political ends;" it was to make it so that federal employees largely stay out of partisan politics. Some rules were loosened for most employees (it used to be that all federal employees were banned from taking an active part in political campaigns, while that's now limited to certain agencies), but running in a partisan campaign is still forbidden. Note that "partisan office" means any candidate is running on behalf of a party; even if the employee is running as an independent, they're breaking the law if one of the other candidates is running as a Democrat or Republican. Obviously, this is the case for the presidency, and you're not going to get every political party to stand down from nominating a candidate.
The exception is for local offices in the DC area (or cities where federal employees are a majority of voters). Because of the massive federal presence there, barring federal employees from local politics could seriously distort local government. Current OPM regulations allow running in local partisan elections in those areas, although the federal employee must themselves be running as an independent. However, the presidency is not a local office, so that campaign would not be "involving the municipality or other political subdivision in which they reside."