This study presents a novel robot-led approach to assessing children's mental wellbeing using a Vision Language Model (VLM). Inspired by the Child Apperception Test (CAT), the social robot NAO presented children with pictorial stimuli to elicit their verbal narratives of the images, which were then evaluated by a VLM in accordance with CAT assessment guidelines. The VLM's assessments were systematically compared to those provided by a trained psychologist. The results reveal that while the VLM demonstrates moderate reliability in identifying cases with no wellbeing concerns, its ability to accurately classify assessments with clinical concern remains limited. Moreover, although the model's performance was generally consistent when prompted with varying demographic factors such as age and gender, a significantly higher false positive rate was observed for girls, indicating potential sensitivity to gender attribute. These findings highlight both the promise and the challenges of integrating VLMs into robot-led assessments of children's wellbeing.