Healthcare workers were the first group scheduled to receive COVID-19 vaccines when they became available in South Africa. Therefore, estimating vaccine confidence levels and intention to receive COVID-19 vaccines among healthcare workers ahead of the national vaccination roll-out was imperative. We conducted an online survey from 4 February to 7 March 2021, to assess vaccine sentiments and COVID-19 vaccine intentions among healthcare staff and students at a tertiary institution in South Africa. We enrolled 1015 participants (74.7% female). Among the participants, 89.5% (confidence interval (CI) 87.2–91.4) were willing to accept a COVID-19 vaccine, 95.4% (CI 93.9–96.6) agreed that vaccines are important for them, 95.4% (CI 93.8–96.6) that vaccines are safe, 97.4% (CI 96.2–98.3) that vaccines are effective, and 96.1% (CI 94.6–97.2) that vaccines are compatible with religion. Log binomial regression revealed statistically significant positive associations between COVID-19 vaccine acceptance and the belief that vaccines are safe (relative risk (RR) 32.2, CI 4.67–221.89), effective (RR 21.4, CI 3.16–145.82), important for children (RR 3.5, CI 1.78–6.99), important for self (RR 18.5, CI 4.78–71.12), or compatible with religion (RR 2.2, CI 1.46–3.78). The vaccine confidence levels of the study respondents were highly positive. Nevertheless, this could be further enhanced by targeted interventions.