Florida law (§922.11) does provide that
The body of the executed person shall be delivered to the medical
examiner for an autopsy. After completion of the autopsy, the body
shall be prepared for burial and, if requested, released to relatives
of the deceased. If a coffin has not been provided by relatives, the
body shall be delivered in a plain coffin. If the body is not claimed
by relatives, it shall be given to physicians who have requested it
for dissection or to be disposed of in the same manner as are bodies
of prisoners dying in the state prison.
Tennessee, on the other hand, is reported to have honored requests to not autopsy and autopsied only 1 of 6 executed prisoners. The Tennessee law (38-7-106) says that
A county medical examiner may perform or order an autopsy on the body
of any person in a case involving a homicide, suspected homicide, a
suicide, a violent, unnatural or suspicious death, an unexpected
apparent natural death in an adult, sudden unexpected infant and child
deaths, deaths believed to represent a threat to public health or
safety, and executed prisoners. When the county medical examiner
decides to order an autopsy, the county medical examiner shall notify
the district attorney general and the chief medical examiner. The
chief medical examiner or the district attorney general may order an
autopsy in such cases on the body of a person in the absence of the
county medical examiner or if the county medical examiner has not
ordered an autopsy. The district attorney general may order an autopsy
in such cases on the body of a person in the absence of the county
medical examiner or the failure of the county medical examiner to act.
The authority ordering the autopsy shall notify the next of kin about
the impending autopsy if the next of kin is known or reasonably
ascertainable. The sheriff or other law enforcement agency of the
jurisdiction shall serve process containing such notice and return
such process within twenty-four (24) hours.
The medical examiner has discretion (autopsy is not an obligation). Louisiana law is a little unclear at the statutory level, because
The coroner shall either view the body or make an investigation into
the cause and manner of death in all cases involving the following...
(2) Sudden or violent deaths...(9) Deaths due to drowning, hanging,
burns, electrocution, gunshot wounds, stabs or cutting, lightning,
starvation, radiation, exposure, alcoholism, addiction, tetanus,
strangulation, suffocation, or smothering... (12) Deaths in prison or
while serving a sentence.
The coroner can discharge that obligation by non-invasive means. Lousiana has a religious-objection provision:
If the family of the deceased objects to an autopsy on religious
grounds, the autopsy shall not be performed unless the coroner finds
that the facts surrounding the death require that an autopsy be
performed in the interest of the public safety, public health, or
public welfare. In such cases the coroner shall provide the family his
written reasons for the necessity of the autopsy.
It's hard to see how there could be a public interest reason compelling an autopsy in the case of a governmental execution, but this is fiction.
Some states have a law providing for religious exceptions to autopsy. You can make a request in S. Carolina, and here is the law in California (moot for the moment since there are no executions in California).
You could try "forum shopping" (changing the state so that you don't get a mandatory autopsy law), but a search of all 50 states is really impractical.