Background Vaccination is an effective way to curtail the burden of COVID-19 in which success depends on a high acceptance of the vaccine. However, addressing concerns among vaccine-hesitant individuals is essential to avoid failure of the immunisation programme. This study sought to assess the concerns and acceptance rates regarding the COVID-19 vaccine among Malaysians. Methods An online questionnaire was distributed to 1411 respondents via a snowball sampling method among Malaysians aged 18 years and above. Results The majority of the respondents were young adults (40.7%), female (62.8%), Malay (63.8%), Muslim (72.3%), married (52.9%), with tertiary education (86.8%) and without medical illness (85%). Social media (97.4%) was the primary source of information regarding COVID-19. The overall acceptance rate was high (83.3%), with the lowest rates among the elderly aged 60 years and above (63.4%) and pensioners (64.6%). Hesitance was caused by concerns regarding side effects (95.8%), safety (84.7%), lack of information (80.9%), effectiveness (63.6%) and religious (20.8%) and cultural factors related to the COVID-19 vaccine (6.8%). Respondents with diabetes mellitus (24.7%) and hypercholesterolemia (23%) were more hesitant to accept the COVID-19 vaccine, at 16.1 and 15.8%, respectively. Predictors of COVID-19 vaccine hesitance were age, religion, and current residence. Conclusions The results indicate a high rate of acceptance of the COVID-19 vaccine among Malaysians. Thus, the Malaysian government and other related agencies should increase their campaign and prepare to implement the COVID-19 mass immunisation programme among Malaysians. However, despite the high acceptance rate, it remains important to address concerns among hesitant individuals by building trust in vaccine safety and effectiveness through adequate information regarding the vaccine.