The details are state-specific, but follow a pattern. Martial law refers to "who has authority". In Washington, the powers of the governor are spelled out here. RCW 38.08.030 says
The governor may by proclamation declare the county or city in which
troops are serving, or any specific portion thereof, to be under
either complete or limited martial law to the extent, in his or her
opinion, that the reestablishment or maintenance of law and order may
"Complete martial law" is the subordination of all civil authority to
"Limited military law" is a partial subordination of civil authority
by the setting up of an additional police power vested in the military
force which shall have the right to try all persons apprehended by it
in such area by a military tribunal or turn such offender over to
civil authorities within five days for further action, during which
time the writ of habeas corpus shall be suspended in behalf of such
The governor shall cause the organized militia of this state at all
times to conform to all federal laws and regulations as are now or may
hereafter from time to time become operative and applicable,
notwithstanding anything in the laws of this state to the contrary.
Except as and when otherwise specifically provided by federal laws,
the organized militia of Washington, or any part thereof, shall be
subject to call for United States service at such times, in such
manner, and in such numbers as may from time to time be prescribed by
the United States.
No provision grants dictatorial power to anyone. The powers of the governor are specified here, and starting here, the law spells out the powers of the governor w.r.t. a "state of emergency", especially here for a list. It does not include postponing or otherwise changing the law regarding elections. (I should point out that Washington has been doing vote-by-mail exclusively, for 15 years, so the virus is a non-issue regarding voting).
It would require a detailed state-by-state study of laws to determine if any state has enacted a provision to the effect that the governor can re-write electoral laws in case of an emergency, but I think it is highly unlikely that any state would grant such power to a governor. Therefore, to change the voting system so that voters in e.g. Ohio must vote by mail or at least drop-box (as in Washington), the legislature would have to change the law (which they did, but just for the primary). Absentee voting is a possible existing work-around: but the voter has to request an absentee ballot.