Disclaimer: Uninformed opinions ahead.
1. Web services (asmx) are old technology now. You should probably go for REST service. As you mentioned that there will be lot of traffic, using JSON as means to share data would be better as it will be smaller in size than traditional messages.
2. If the entities are unrelated or there are no complex relationships, you might want to consider micro services based architecture. This is more or less inline with what you mentioned: Each entity with its own service. Entity here should rather be a business entity than the class which you would create.
3. Assuming less relationships, it will be easy for you to figure out aggregates i.e. a set of data that can independently represent some information. If this applies, NoSQL would help.
No, I am not saying ditch MS technologies and go for MEAN stack (while it would be an option here). Even a micro-service based architecture with multiple SQL databases would be fine.
So, IMHO, here is what might work out:
Web client -> HTTP Get/Post/so on -> REST resource (micro service) -> database and back.
In parallel, you might need to synchronize databases which can be done using technology of your choice.