I need to make a Rest call to create an entry for on a list when a button on a page is clicked. This works as long as the user has read and write access. However, if a user only has read access I get 'Access Denied' on my rest call. Is there anyway around this?

I should not that this button is on a page layout on the root site, this call is not being made from an app.

REST calls use the permissions of the user or the App. If it was otherwise then a user could inject JS into a page and do anything. If you are working on premise, then you create your own web service that has the write permission and then call that.

That's a very good point. The more I thought about this the more I realized that if I could do this through jquery, it would be a huge security breach. Thanks!