Oral Administration of Immunizations
First, ongoing inflammatory process/immune reaction in the gastrointestinal tract is not necessary associated with vomiting or diarrhoea. As an example we can consider infection with Helicobacter Pyroli, which may last for many years unnoticed. 

Second, exposition to foreign antigens obtained form food or microflora and subsequent antigen presentation to the immune cells (which is indeed partially driven by locally released cytokines) occur in the GI tract all the time and again, is not usually manifested by vomiting or diarrhoea. 

But moving to vaccines, attenuated oral vaccines (like Polio Sabin or Rotavirus) are developed in order to diminish the risk of pathological symptoms. Attenuation weaks the pathogen that is usually no longer able to proliferate and invade, but still poses antigens that may be presented to the immune cells and induce long-term immunological memory. Moreover, immune response (antigen presentation, production of the local IgA, B-cells affinity maturation) triggered by attenuated vaccines is the closest to the one induced by full infection (in comparison to inactivated/toxoid vaccines), but significantly weaker and localised. This we mean by "mimicking a natural infection".  

Also, see: https://www.sciencedirect.com/science/article/pii/S0042682215001725