Please read the following Microsoft document:
Custom storage providers for ASP.NET Core Identity
See the diagram in the above document and description blow:
"To create a custom storage provider, create the data source, the data access layer, and the store classes that interact with this data access layer (the green and grey boxes in the diagram above). You don't need to customize the managers or your app code that interacts with them (the blue boxes above)."
You will be able to use your Web API as the Data Source (the gray box shown in the diagram).
Create custom Data Access Layer and Identity Store (the green boxes in the diagram). Samples are provided in the following pages:
Using your own database schema and classes with ASP.NET Core Identity and Entity Framework Core [http://danderson.io/posts/using-your-own-database-schema-and-classes-with-asp-net-core-identity-and-entity-framework-core/]
Customize ASP.NET Core Identity
The Data Access Layer in the above samples use the Entity Framework Core. Therefore, please rewrite it so that the Identity Store layer can access to your Web API.
The UserManager and SignInManager in the existing Identity Manager layer (the blue box in the diagram) in the ASP.NET Core Identity will interact with the methods in the custom Identity Source.