You can collect VMware ESXi system logs by using the vSphere Client or vSphere Web Client. For details on the collection methods, visit the VMware official knowledge base (KB):

Collecting diagnostic information for VMware ESX/ESXi (653)

You can also collect VMware ESXi system logs by using vm-support commands. For details on the collection methods, visit the VMware official KB:

"vm-support" command in ESX/ESXi to collect diagnostic information (1010705)