Is it possible that a different orthodontist can fix my permanent retainers?

Yes: If you cannot or do not want to return to the original orthodontist another one can repair or replace your retainer.

Possible: It is very likely that another orthodontist can help you with the orthodontic appliance. Every time a doctor sees a new patient, it is important that s/he performs examination first so as not to overlook something that is more serious. You are likely to have an exam, then your new dr may ask for your clinical note/chart from your previous dr.

Maybe: Some orthodontists don't believe in fixed retainers. Most "permanent" retainers are simply bonded to the lingual surfaces of the teeth and many times, do break away. If this happens multiple times, i suggest removable devices.

Yes, but ..: If you still live near the first orthodontist, it is best, if possible to continue care with the same provider.

Yes: Best to return to treating Orthodontist. However if that's not possible, see another Orthodontist for care.