Background: COVID-19 vaccine acceptance by the nurses of the future is important to increase the population immunity. Aim: This study aimed to determine the attitudes and concerns of nursing students towards the coronavirus disease-2019 (COVID-19) vaccine, their willingness to be vaccinated, and the factors affecting their willingness in the second wave of the COVID-19 pandemic. Method: This cross-sectional study was conducted between February 1, 2021, to March 30, 2021, using an online questionnaire in 498 nursing students in Turkey. Results: While 64.5% of nursing students had the intention to be vaccinated against COVID-19, 35.5% had no intention to be vaccinated (29.3% hesitant and 6.2% unwilling). Students expressed their attitudes and concerns on the lack of COVID-19 vaccine information (65.7%), its effectiveness (41.6%), safety (45.8%), and side effects (50.8%). Students did not intend to be vaccinated due to insufficient trust in the vaccine effectiveness (84%), the continuous COVID-19 mutation, vaccines side effects (68.9%), and not trusting the supplied vaccine (55.4%). Students’ high education level, high family income perception, previous vaccine rejection, family members with COVID-19 infection, COVID-19 information from official institutions, trusting health professionals’ COVID-19 explanations, and attitudes and concerns toward vaccines for their intention to be vaccinated are the affecting factors (p < 0.05). Conclusions: Nursing students are known to be future healthcare professionals and play a decisive role in counseling individuals in the community on the risks of COVID-19 and the benefits of the vaccine. Therefore, focusing on training that is aimed at increasing vaccine knowledge, eliminating their negative attitudes and concerns, and building confidence in vaccines is necessary.