Using dapper.net and repository pattern in Azure websites
Always program to abstraction, meaning it is always preferred to write your code against interface or abstract class rather than specific implementation.  So, IStoreRepository  would be a good idea.
If you are not going to ever change your implementation, do not use abstract or interfaces. Make it simple and readable. 
If you are considering using mapper and all, use DI container too so that you don't have to do this:

var db = new StoreRepository();
// creating instance like this is not a good idea, rather inject the dependencies 


Same thing goes for your SQL helper class too.  Inject as a dependency in your repository.
If you use EF/nhibernate, then your code will not be repeating as all code will have the same kind of code.  You write a generic base class for them and pass the type of data you want to retrieve it from the database.  But in case of hand-written SQL, that could not be done that easily as you need to write different SQL for different type. Again, that is your choice.
It is good to use async await for managing an SQL connection so you don't just block the currently-working thread. Overhead is small in overall picture.