How to enable

Managed Identities

for Azure API Management services? Required in order to enable custom

hostnameConfiguration

const service = new azure.apimanagement.Service(`${prefix}-service`, {
  resourceGroupName: resourceGroup.name,
  sku: {
    name: 'Developer',
    capacity: 1
  },
  publisherName: 'Bob',
  publisherEmail: 'alex.b@d.c',
  hostnameConfiguration: {
    proxies: [{
      hostName: domainName,
      // certificate: '',
      // certificatePassword: '',
      keyVaultId: pulumi.interpolate`${keyVault.vaultUri}certificates/generated-cert/90951e125ada483c8a1e76440fa484c7`,

      negotiateClientCertificate: false,
      defaultSslBinding: true
    }]
  }
})

Have you tried

identity: {
        type: "SystemAssigned",
    },

?

that did the trick, thanks Mikhail