Ugh, no, you don't want to do this.
You can, by allowing the "group" called "anyone" the "issue create" permission in the permission scheme, but you will end up in a mess because you can't track them. And you certainly can't put such a system on the internet - it's only going to last a few hours before drowning in spam.
If you really need to do something like this, then get JIRA Service Desk instead. You still have to create accounts for the users, but "customer" accounts do not cost anything, and you'll be able to track them properly.
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