This paper introduces an online inspection algorithm that enables an autonomous UAV to fly around a transmission tower and obtain detailed inspection images without a prior map of the tower. Our algorithm relies on camera-LiDAR sensor fusion for online detection and localization of insulators. In particular, the algorithm is based on insulator detection using a convolutional neural network, projection of LiDAR points onto the image, and filtering them using the bounding boxes. The detection pipeline is coupled with several proposed insulator localization methods based on DBSCAN, RANSAC, and PCA algorithms. The performance of the proposed online inspection algorithm and camera-LiDAR sensor fusion pipeline is demonstrated through simulation and real-world flights. In simulation, we showed that our single-flight inspection strategy can save up to 24 % of total inspection time, compared to the two-flight strategy of scanning the tower and afterwards visiting the inspection waypoints in the optimal way. In a real-world experiment, the best performing proposed method achieves a mean horizontal and vertical localization error for the insulator of 0.16 +- 0.08 m and 0.16 +- 0.11 m, respectively. Compared to the most relevant approach, the proposed method achieves more than an order of magnitude lower variance in horizontal insulator localization error.