I wasn't able to find any helpful information in the logs that could help identify the cause. Interestingly enough, enabling the service through SSH worked. Here's how I did it. First, you'll need to connect to your host via SSH (you will need to enable the SSH daemon first).
Once connected, type in "esxcli system snmp get" to see the current snmp configuration.
To enable SNMP, we first need to configure the community strings. To do that, type in "esxcli system snmp get --communities=<community strings>". Then, enable the service by typing in "esxcli snmp set --enable=true".
You can then verify the new configure by typing in "esxcli system snmp get" again.
Now refresh the vSphere Host Client, and the service should be running.
