At this point the MySQL extension is deprecated meaning that it still works, but in the future, it will be removed.
The MySQL library is still available for the host to enable. If they are not giving a previous version option, they should be able to enable the MySQl library for the time being. The Servers error reporting may be set to a level that will show deprecation errors you may have to edit the php.ini file to tell it to hide deprecation errors in the error_reporting setting.
Currently we are working on eCart 6, once that is finished, we will be working on Data Bridge to upgrade the database connections scripts to use the MySQLi library.