Efficient Random Access (RA) is critical for enabling reliable communication in Industrial Internet of Things (IIoT) networks. Herein, we propose a deep reinforcement learning based distributed RA scheme, entitled Neural Network-Based Bandit (NNBB), for the IIoT alarm scenario. In such a scenario, the devices may detect a common critical event, and the goal is to ensure the alarm information is delivered successfully from at least one device. The proposed NNBB scheme is implemented at each device, where it trains itself online and establishes implicit inter-device coordination to achieve the common goal. Devices can transmit simultaneously on multiple orthogonal channels and each possible transmission pattern constitutes a possible action for the NNBB, which uses a deep neural network to determine the action. Our simulation results show that as the number of devices in the network increases, so does the performance gain of the NNBB compared to the Multi-Armed Bandit (MAB) RA benchmark. For instance, NNBB experiences a 7% success rate drop when there are four channels and the number of devices increases from 10 to 60, while MAB faces a 25% drop.