Yes, Microsoft Azure OpenAI Service provides REST APIs and SDKs (like Python) to interact with generative models, including image generation (like DALL·E through OpenAI integration). See: The Azure AI Foundry SDK
Consistency in character and theme is challenging because generative models typically work statelessly. However, you can for example write an agent using Azure AI Agent Service that has a system prompt that includes your general / character description. Then you only need to send the description for the current picture as a user message. But I would deal with it and always send a complete prompt when generating the image.