Zabbix supports API usage:
https://www.zabbix.com/integrations/meraki
If there's something that Zabbix doesn't support, this should be relatively easy to integrate, as the Meraki API is REST based
https://developer.cisco.com/meraki/api-v1/get-network-events/ would be a good place to start
Also, it would be helpful to understand what you means by "logs"