Is there a way that Jira can set due dates for issues atomatically based on the milestone dates that they are tied to?
No. Mostly.
I'm assuming by "milestone" you mean a Jira version. Two problems - first, an issue can be tagged with more than one version, so you'd need to think about how you handle that. Second is that you're going to need to find or write a bit of code to implement this facility. I can't see it being horrendously hard, but you've said you're using OnDemand, so you won't be able to install whatever you write or find.
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