There is a permission within the permission scheme for resolving issues.
Make sure they are assigned to a role granted that permission in ther permission scheme used by the projec.t
https://confluence.atlassian.com/display/JIRA/Managing+Project+Permissions
Just to add on, you can use the JIRA Admin Helper tool to quick check the user's permission on specific issue/project. Refer to the following page:
https://confluence.atlassian.com/display/JIRA/JIRA+Admin+Helper
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