i suspect this is a poorly written story, missing is 'where' the driver
was when the blood was drawn - guessing that they took her to hospital
or the station. i don't know of 'any' jurisdiction where a blood draw is
performed at the location of the infraction.

from the story, they would have had probable cause to take her to
another location

see south dakota law:

32-23-10.

Operation of vehicle as consent to withdrawal of bodily substances and
chemical analysis Submission to withdrawal or analysis following arrest.

Any person who operates any vehicle in this state is considered to have
given consent to the withdrawal of blood or other bodily substance and
chemical analysis of the person's blood, breath, or other bodily
substance to determine the amount of alcohol in the person's blood and
to determine the presence of marijuana or any controlled drug or
substance or any substance ingested, inhaled, or otherwise taken into
the body as prohibited by § 22-42-15 or any other substance that may
render a person incapable of safely driving. The arresting law
enforcement officer may, subsequent to the arrest of any operator for a
violation of § 32-23-1, require the operator to submit to the withdrawal
of blood or other bodily substances as evidence.

32-23-10.1.

  Refusal to submit to chemical test or allow withdrawal of bodily
substance admissible into evidence.

If a person refuses to submit to chemical analysis of the person's
blood, urine, breath, or other bodily substance, or allow the withdrawal
of blood or other bodily substance for chemical analysis as provided in
§ 32-23-10, and that person subsequently stands trial for violation of §
32-23-1 or § 32-23-21, such refusal may be admissible into evidence at
the trial.

32-23-14.

  Persons authorized to withdraw blood for test Other bodily
substances--Liability of person administering test.

Only a physician, laboratory technician, registered nurse, physician's
assistant, phlebotomist, expanded role licensed practical nurse, medical
technician, or medical technologist may withdraw blood for the purpose
of determining the alcoholic content therein. This limitation does not
apply to the taking of a breath or other bodily substance specimen. Such
authorized persons, acting on the presumption of consent in § 32-23-10,
and any hospital or facility employing such persons, are not liable and
may not be held to pay damages to the party from whom the blood sample
is withdrawn, if the withdrawal is administered with usual and ordinary
care. No person authorized to withdraw blood under this section may be
required or forced to withdraw blood for the purposes outlined in this
chapter, unless required pursuant to a written agreement.