Actually, it has little to do with MRTG, as I quickly established that snmptable, snmpwalk, and snmpget all suffered from the same problem: Even though I had properly configured my Airport Extreme to allow SNMP access, I'd get back an error that said "Timeout: No Response from [ip address]".
I tried several settings, and each time I clicked "Update" in the Airport Admin Utility on my Mac, it said the router restarted (with a surprisingly quick restart time) and the setting appeared to have been applied.
Still inaccessible at port 161, I figured that the Airport Extreme probably needed a hard restart. Tsk tsk to Apple for allowing a quick soft restart in the SNMP settings window to let the user think the setting had been applied when it hadn't! If you're running commands like
snmpwalk -v 1 -c public [Airport IP Address] system
and getting SNMP "Timeout: No Response" errors, even though you've clicked the "Allow SNMP" checkbox in the "Logging and Statistics" tab of the Advanced panel in Airport Utility, the solution is simple.
snmpget -v 2c -c public -M /usr/share/snmp/mibs:$HOME/share/mibs \
-m+AIRPORT-BASESTATION-3-MIB [Airport IP Address]
Right click (control click) your Airport Extreme in Airport Utility and click the "Restart..." option. It should take about a minute to reload, and you'll notice your snmpwalk/snmpgets are now succeeding, your Airport Extreme will be accepting UDP packets on port 161, and software like MRTG (if properly configured) will start to work.