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Figured it out. For anyone else wondering, here’s how to do this:

var sharedSubscriptionProvider = new Provider(sharedSubscriptionName, new ProviderArgs() { SubscriptionId = sharedSubscriptionId });

var keyVault =
	Output.Create(GetVault.InvokeAsync(
		new GetVaultArgs()
		{
			ResourceGroupName = sharedResourceGroupName,
			VaultName = sharedKeyVaultName
		},
		new InvokeOptions()
		{
			Provider = sharedSubscriptionProvider
		}));