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I have deployed a regression model on Azure ML, is it possible to get the model weights/coefficients of the model programmatically from Azure, rather than getting predicted value?

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No, currently Microsoft  does not feature the exporting weights within the models including the Azure Machine Learning.

If you have a method for extracting weights from Python models, you may be able to work this out using the execute Python Script module.

The primary purpose of Azure Machine Learning is to make deployable and scalable web services from the machine learning modules. Though the authoring experience for creating ML models is great, it is not intended to be a place to create and export models, but instead a place to create and operationalize your models.

UPDATE New features may make this answer outdated.

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