My webhost has told me that, "mod_rewrite is not allowed on our server due to some
security risks." Can I run Artio joomsef without mod_rewrite"?

Hello,

you are perfectly correct. Running JoomSEF or any other Joomla based SEF component without Apache's mod_rewrite is not possible. The story about security issue is just nonsense to me.

The only recommendation I can give is to change your webhosting provider. We should also be able to help you if you need.

Best regards,
Jirka