The iconv extension is required in Magento. Is it possible to enable it?

It's absolutely possible to enable iconv.

The Magento help forum gives the command that you can run to enable missing php extensions, including iconv:

sudo apt-get install php5-intl php5-iconv

As the example uses php5, make sure you replace that with the version of php you have installed on your server.

Enable iconv in Magento

CPanel's help forums also gives a simple way of accomplishing this by using 'yum install. The syntax of the command as is follows:

yum install ea-php56-php-iconv

As noted in the previous example, make sure you use your version of php in place of 'php56'

Enable inconv for CPanel