Background: Access to healthcare for young people is essential to ensure they can build a foundation for a healthy life. However, during the COVID-19 pandemic, many people avoided seeking healthcare, adversely affecting population health. We investigated the factors associated with the avoidance of healthcare for Australian young people when they reported that they needed healthcare. We were able to compare healthcare avoidance during the COVID-19 pandemic with healthcare avoidance prior to COVID-19. Methods: We used two recent data collection waves from the Longitudinal Study of Australian Children (LSAC)—Wave 9C1 during the COVID-19 pandemic in 2020, and Wave 8 data which were collected in 2018. The primary outcome of this study revealed the avoidance of healthcare among those who perceived the need for care. Bivariate analyses and multiple logistic regression models were employed to identify the factors associated with the avoidance of healthcare during the COVID-19 and pre-COVID-19 periods. Results: In the sample of 1110 young people, 39.6% avoided healthcare during the first year of the COVID-19 pandemic even though they perceived that they had a health problem that required healthcare. This healthcare avoidance was similar to the healthcare avoidance in the pre-COVID-19 pandemic period (41.4%). The factors most strongly associated with healthcare avoidance during the COVID-19 pandemic were female gender, an ongoing medical condition, and moderately high psychological distress. In comparison, prior to the pandemic, the factor associated with healthcare avoidance was only psychological distress. The most common reason for not seeking healthcare was thinking that the problem would spontaneously resolve itself (55.9% during COVID-19 vs. 35.7% pre-COVID-19 pandemic). Conclusions: A large proportion of youths avoided healthcare when they felt they needed to seek care, both during and before the COVID-19 pandemic.