The activity log doesn't have it's own level of permissions. If you are referring to the activity tab that you see on an issue itself in Jira, then as long as the user can browse the issue, they should have access to that resource as well.
If they are not seeing any results there, then it's likely there is something wrong with the way the page is loading. Since this element can contain potentially a lot of data, since it's tracking all the changes to an issue, I don't think this is a permissions problem. Just feels like it might be a performance problem, such as the resource is timing out before being able to request back all the data. Is your user seeing some kind of message that indicates this is permissions based? I would be interested to see what that looks like here if that is the case.
Otherwise, it might be better to create a Cloud support case for this issue over at https://support.atlassian.com select Technical problem as the case type and be sure to enter your Cloud URL when selecting the product. This will make sure your request is routed to our Cloud support team that can then investigate your specific Cloud site better.
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