Organic neuromorphic devices can accelerate neural networks and integrate with biological systems. Devices based on the biocompatible and conductive polymer PEDOT:PSS are fast, require low amounts of energy and perform well in crossbar simulations. However, parasitic electrochemical reactions lead to self-discharge and the fading of learned conductance states over time. This limits a neural network's operating time and requires complex compensation mechanisms. Spiking neural networks take inspiration from biology to implement local and always-on learning. We show that these spiking neural networks can function on organic neuromorphic hardware and compensate for self-discharge by continuously relearning and reinforcing forgotten states. In this work, we use a high-resolution charge transport model to describe the behavior of organic neuromorphic devices and create a computationally-efficient surrogate model. By integrating the surrogate model into a Brian 2 simulation, we can describe the behavior of spiking neural networks on organic neuromorphic hardware. A biologically-plausible two-layer network for recognizing 28x28 pixel MNIST images is trained and observed during self-discharge. The network achieves, for its size, competitive recognition results up to 82.5%. Building the network with forgetful devices yields superior accuracy during training with 84.5% compared to ideal devices. However, the trained networks without active spike-timing-dependent plasticity quickly lose their predictive performance. We show that online learning can keep the performance at a steady level close to the initial accuracy, even for idle rates of up to 90%. This performance is maintained when the output neuron's labels are not revalidated for up to 24 hours. These findings reconfirm the potential of organic neuromorphic devices for brain-inspired computing. Their biocompatibility and the demonstrated adaptability to spiking neural networks open the path toward close integration with multi-electrode arrays, drug-delivery devices, and other bio-interfacing systems as either full organic or hybrid organic-inorganic systems.