Some states (including big ones like California and NY and conservative ones like Mississippi and West Virginia) have no exemptions to school vaccine requirements other than medical exemptions.
California used to have more exemptions and they were used heavily by new age/hippie types (i.e. not conservatives) to skirt vaccination requirements so the laws were changed recently.
Since the change there have been a number of doctors willing to grant medical exemptions to any who want them. [0] It isn't as much of an impediment as it appears.
The article you linked talks about a new law that invalidated those exemptions from that doctor because they were not supported by clinical guidelines.
It's just an example. There are lots more doctors who will give exemptions just to make parents happy. There is clearly friction between parents/doctors/state over this policy.
It's an example that you provided a source that says the exact opposite of your example. In California there is a new law designed to specifically combat your example and it's been quite effective.