Usually each site has a separate tmpfs, mounted on /opt/omd/sites/SITENAME/tmp. You can check the config with omd config show TMPFS as site user.
But even if you don’t use a (fast because RAM based) tmpfs, in any case you need the ~/tmp directory in the site home. By default it has 0755 permissions (drwxr-xr-x) and is owned by the site user/group. Check if perhaps this directory is missing in your site’s home or has wrong permissions/owner/group.