Is it possible to train this image classifier?
For sure, is a really feasible application! Indeed, there does exist Deep Learning models for that.

I suggest you to start by doing transfer learning on a pre-trained model like this: https://github.com/satojkovic/DeepLogo Typically you remove the last layers of the network and add layers that adapt to your data. Then, train these last new layers with your data (freezing the layers you have kept from the original model).

Another good advice is to perform data augmentantion on your dataset to include ligth, scale and rotation variants of your images.

Summary:


It's a feasible task
Check out pre-build models
Learn about transfer learning on pre-trained models
Learn about data augmentation to include robustness against different scenarios and env. conditions