Does poll not allow multiple anonymous votes?
The problem is that the poll module only allows one vote per anonymous IP address.  In the case of your hosting provider I suspect that they are using some sort of proxy that sends all requests to Drupal from the same IP address.

The Poll Enhancements Module modifies this behavior to allow multiple anonymous votes from the same IP address.  The documentation on this module's description page also suggests the Poll Improved Module may solve this as well, but it does not have a production release at this time.