I'm gonna answer your question, hoping that your elmah.axd is under heavy authentication :)
you can setup an akamai delivery configuration to avoid caching any resource you decide, you can specify a rule on the property manager where you can match by path or extension or whatever other combination/condition and apply a no-store behaviour, this will basically bypass the cache for every hit and akamai will just proxy the client's call to your servers, still effectively hiding the origin but actually querying it every time
PS. if i were you i would completely close-off the elmah script using akamai (creating a rule that will just redirect or bounce any requests to it) and access only calling the origin (which should be known only to a handful of people)
PPS. a good practice tip: don't allow random query strings to create new cache-keys and bypass akamai's cache, you are making your origin server potentially vulnerable to flood, rendering the nice akamai shield less effective