Hi,
AspNet Zero doesn't support only the API Key at the moment because each action must be related to a user. You can add a new field to user entity (API key) or create a separate table to related API Keys with Users. Then, when a reuqest is sent to server, you can get the API Key and find the related user and generate a token.
For your custom domain resolver issue, you can take a look at https://github.com/aspnetzero/aspnet-zero-core/tree/dev/angular/src/shared/multi-tenancy/tenant-resolvers. We implement a similar approach on our angular app. Angular app retrieves the TenantId from the current URL (subdomain or query string etc...) and sends Abp.TenantId to API with every request. You can implement a similar approach for your VueJS app.
That GitHub link did not work. I get a 404