Can you a export data-tier task in SQL Server 2016 Management Studio?

Yes, mostly. There could be instances when you may not be able to. It appears that your database must be compatible with Microsoft Azure SQL Database v12. There could be other instances as well. Please refer to the documentation.

Herein are screen shots of how you export a data-tier application from SQL Server 2012. You can of course connect to SQL Server 2012 from SQL Server 2016 Management Studio.

The instance of SQL Server 2012 has a database called pubs, one of the oldest of Microsoft's sample databases. This will be used in the demonstration. You can invoke a wizard to do this task from inside the SQL Server 2016 Managment Studio as shown.

This pops-up the wizard as shown.

The steps are clearly indicated. Click Next.

You will be creating a .bacpac file. Provide a name and it usually saves the file to the default location. After providing a name by browsing to the default location, click Next.

All the data also goes with it and you can deselect some if you want to. Click Next.

Here is a summary of actions that will be taken. Click Finish.

There is a progress stage when the action is processed as shown.

After processing, if it succeeds you should see the following screen.

The task succeeds and a .bacpac file is created.

Follow this post to review a case when the task files.

Read these popular blog posts related to data-tier applications:
http://hodentekmsss.blogspot.com/2010/07/sql-server-2008-r2-and-data-tier.html
http://hodentekmsss.blogspot.com/2013/12/deploying-dac-to-sql-server-2014.html

You may want to read this as well:
http://hodentekhelp.blogspot.com/2016/08/what-is-easy-way-to-move-database-from.html