Authentication cookies are the standard for browser-based sessions (like a standard ASP.NET Core MVC or Blazor Server app). JWTs are typically used in distributed scenarios where a remote service needs to trust the client application via a signature, often when the client is a separate SPA or mobile app.
To give you the best advice, we need to know your architectural intentions: Is this Web API shared by other applications?
If the Web API only supports the Blazor Server application: You likely do not need a Web API at all. Since Blazor Server code runs on the server, you can simply inject your logic/data services directly into your components. The user is already authenticated via the standard cookie, so you can handle authorization logic right there in the Blazor app without making an extra HTTP call.
If the Web API is shared between multiple applications: You will need a design change. You need a centralized way for different clients (Blazor, Mobile, etc.) to authenticate and obtain a token. This typically involves using an OpenID Connect provider (a token server) that both the Blazor app and the Web API trust.
If you must call a local API from Blazor Server (sometimes done for consistency), you have to manually copy the authentication cookies from the browser request to the HttpClient request, which is tedious. Direct service injection is much preferred.