In Bulgaria, vaccination coverage against the SARS-CoV-2 virus is low. The reasons for this fact are many and varied. The aim of the present study was to establish what the attitudes towards the COVID-19 vaccination process are among students from various specialties from several Bulgarian universities. In this research, 600 students participated, divided into two groups: Doctor of Medicine (MD) students (n = 300) and non-MD students, i.e., students of specialties, such as mathematics, engineering, finance and economics, law, human sciences, etc. (n = 300). Each respondent completed a questionnaire which was divided into three parts with closed questions. The mean age of all students was 21.19 ± 1.87 years (95% CI: 20.48–21.90). The female sex dominated among the analyzed participants (sex ratio: female/male = 1/0.85). Nearly 62% (371/600) of individuals declared that they have been COVID-19 vaccinated with at least one dose (p < 0.001). Overall, 33% of the participants sought information on vaccines from video sharing platforms and 36.0% (216/600) from social media platforms. From the conducted multivariable logistic regression the odds of vaccination against COVID-19 were 6.225 times higher in individuals with a positive attitude towards these vaccines than in people with a negative attitude towards them (p < 0.001). We have found that those students who trust the international health organizations had an OR of 2.365 (p = 0.004) to be SARS-CoV-2 vaccinated. We estimated that the odds of vaccination against SARS-CoV-2 among children were 4.794 times higher in parents (students) who had been vaccinated than in non-vaccinated parents (students) (p < 0.001). Our results could support the national public health organizations, the national educational/scientific systems, and the management of Bulgarian universities in making future decisions about the field of COVID-19 control and prevention.