Delayed and partially observable state information poses significant challenges for reinforcement learning (RL)-based control in real-world autonomous driving. In highway on-ramp merging, a roadside unit (RSU) can sense nearby traffic, perform edge perception, and transmit state estimates to the ego vehicle over vehicle-to-infrastructure (V2I) links. With recent advancements in intelligent transportation infrastructure and edge computing, such RSU-assisted perception is increasingly realistic and already deployed in modern connected roadway systems. However, edge processing time and wireless transmission can introduce stochastic V2I communication delays, violating the Markov assumption and substantially degrading control performance. In this work, we propose DAROM, a Delay-Aware Reinforcement Learning framework for On-ramp Merging that is robust to stochastic delays. We model the problem as a random delay Markov decision process (RDMDP) and develop a unified RL agent for joint longitudinal and lateral control. To recover a Markovian representation under delayed observations, we introduce a Delay-Aware Encoder that conditions on delayed observations, masked action histories, and observed delay magnitude to infer the current latent state. We further integrate a physics-based safety controller to reduce collision risk during merging. Experiments in the Simulation of Urban MObility (SUMO) simulator using real-world traffic data from the Next Generation Simulation (NGSIM) dataset demonstrate that DAROM consistently outperforms standard RL baselines across traffic densities. In particular, the gated recurrent unit (GRU)-based encoder achieves over 99% success in high-density traffic with random V2I delays of up to 2.0 seconds.