This FAQ attempts to answer the question "can vaccines cause disease" by saying "No" but the problem is it can't.

This is done in three parts:
Part 1: "Can vaccines cause “shedding”? (can they cause or spread the diseases they are meant to protect you from?)
The short answer:
No, with some extremely rare exceptions"

If the answer it not "No." and is instead "no, but", then the answer is "yes". Yes vaccines can cause the disease they are supposed to protect against. The report then goes on to give examples of where it happened while claiming it can't.

Part II: The longer answer:

In order for a disease to be spread;

a person has to be infected with a germ

the germ has to be shed
the shed germ has to be transmitted
the germ has to be virulent enough to cause disease

There is a theoretical risk that live attentuated [sic] vaccines like the MMR can be found in body fluids (known as shedding), but that does NOT mean they are capable of causing the disease in the person vaccinated, much less being passed on, or causing disease in anyone else.

Except it did happen here in 2010 "We demonstrated excretion of the Schwarz measles vaccine virus in a child with a vaccine-associated febrile rash illness in urine and in pharyngeal excretions."

"Virus excretion in vaccinees has been reported before [8-10], but to our knowledge, this is documented for the first time for the Schwarz vaccine strain.
https://www.eurosurveillance.org/content/10.2807/ese.15.35.19652-en

and here in 2013:

"In this report we describe a case of measles-mumps-rubella (MMR) vaccine-associated measles illness that was positive by both PCR and IgM, five weeks after administration of the MMR vaccine. Based on our literature review, we believe this is the first such case report which has implications for both public health follow-up of measles cases and vaccine safety surveillance."
Case of vaccine-associated measles five weeks post-immunisation, British Columbia, Canada, October 2013
Eurosurveillance / Volume 18, Issue 49, 05/Dec/2013
https://www.eurosurveillance.org/content/10.2807/1560-7917.ES2013.18.49.20649

And when it did happen where in 2013 and somebody died:

We describe a death in a 15-mo-old girl who developed a varicella-like rash 20 d after varicella vaccination that lasted for 2 mo despite acyclovir treatment. The rash was confirmed to be due to vaccine-strain varicella-zoster virus (VZV). This is the first case of fatal varicella due to vaccine-strain VZV reported from the United States.
Fatal varicella due to the vaccine-strain varicella-zoster virus
Hum Vaccin Immunother. 2014 Jan 1; 10(1): 146–149.
Published online 2013 Aug 27. doi: 10.4161/hv.26200
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4181020/

This is not "rare":

"13 out of the 16 immunized measles cases had measles from the vaccine strain
[Rapid identification of Beijing measles vaccine virus and wild virus by multiplex real-time fluorescent PCR].
Chen M, et al. Zhonghua Yu Fang Yi Xue Za Zhi. 2012.
https://www.ncbi.nlm.nih.gov/m/pubmed/23363872/

Part III "The even longer answer! [...] The following are live attenuated vaccines: Measles, Mumps, Rubella, Chicken pox, Rotavirus, Nasal flu vaccine (Flumist), Oral Polio vaccine."

All of these are capable of causing the disease they're supposed to protect against.