The rapid increase of space assets represented by small satellites in low Earth orbit can enable ubiquitous digital services for everyone. However, due to the dynamic space environment, numerous space objects, complex atmospheric conditions, and unexpected events can easily introduce adverse conditions affecting space safety, operations, and sustainability of the outer space environment. This challenge calls for responsive, effective satellite object detection (SOD) solutions that allow a small satellite to assess and respond to collision risks, with the consideration of constrained resources on a small satellite platform. This paper discusses the SOD tasks and onboard deep learning (DL) approach to the tasks. Two new DL models are proposed, called GELAN-ViT and GELAN-RepViT, which incorporate vision transformer (ViT) into the Generalized Efficient Layer Aggregation Network (GELAN) architecture and address limitations by separating the convolutional neural network and ViT paths. These models outperform the state-of-the-art YOLOv9-t in terms of mean average precision (mAP) and computational costs. On the SOD dataset, our proposed models can achieve around 95% mAP50 with giga-floating point operations (GFLOPs) reduced by over 5.0. On the VOC 2012 dataset, they can achieve $\geq$ 60.7% mAP50 with GFLOPs reduced by over 5.2.