Housing instability is a widespread phenomenon in the United States. In combination with other social determinants of health, housing instability affects children's overall health and development. Drawing on data from the 2022 National Survey of Children's Health, we employed multiple logistic regression models to understand how sociodemographic factors, especially housing instability, affect mental health outcomes and treatment access for youth aged 6-17 years. Our results show that youth facing housing instability have a higher likelihood of experiencing anxiety (OR: 1.42, p<0.001) and depression (OR: 1.57, p<0.001). Furthermore, youth experiencing both mental health conditions and housing instability are significantly less likely to receive mental health services in the past year, indicating the substantial barriers they face in accessing mental health care. Based on our findings, we highlight opportunities for digital mental health interventions to provide children experiencing housing instability with more accessible and consistent mental health services.