Steve has written a detailed account of the process of getting the required certifications and having them apostilled in the comments sections of Bringing In Animals. That's exactly what we did. We were lucky that our vet was an APHIS vet so that I didn't have to go through the extra stage--which a friend of ours had to--of getting the certificate from (in our case) Olympia, which would have added a minimum of at least one day to our process.
These days, you'd have to live in a really out-of-the-way place for your vet not to be aware of what it takes to ship animals internationally. Always a good place to start.
But as I added in the Comments section and wish to emphasize: the rules for bringing in animals may have changed. ALWAYS check with the Panamanian Consulate or Embassy, always. Make no assumptions.
Also, for those who may be interested, Steve has given José Saenz's email address. Again, you can do it on your own, but especially if you are bringing in animals in pet cargo, it is advisable to go through an agent.
I should mention, too, that if you arrive when the official vet is not there, your animals will go into quarantine overnight at the airport, for which you will pay (I have no idea how much). Those with whom I've talked who had had to go this route say that the people in charge are kind and courteous, and helpful. Panamanians love animals. Steve's comment about drawing a crowd because of their dog's unusual breed is right on.
As for Eugene Malek International Airport in David: yes, they are lengthening the runways so they can use larger planes, but I have no idea when that's going to happen. Perhaps they will then add animal immigration but I doubt it. The airport itself is small, and while it's touted as being 'International", it just means routine flights entering from Costa Rica and I think, at this time, Columbia, but wouldn't swear to it. My guess is that the Panamanian government will still funnel all formal immigration through Tocumen for the foreseeable future. With immigration to Panamá from the US (and quite possibly soon from western Europe) dropping off, they don't have a lot of reason to expand services to David.
All great information. Please send any additional, and I'll post it.