Hi Al – thanks for posting your inquiry here. There are a number of resources you may find helpful in locating a short-term rental.
If you scroll down to the bottom of this page you’ll find numerous realtors listed, who may be able to help you secure a rental: http://internationalliving.com/country_clubs/panama/rolodex/
VRBO (http://www.vrbo.com/vacation-rentals/central-america/panama) often has short-term listings for rooms, apartments, and more.
Airbnb is another site that has short-term listings, here https://www.airbnb.com/s/Panama–Panama
TripAdvisor has a forum where you may post an inquiry, as well as lots of listings, here: http://www.tripadvisor.com/ShowForum-g38273-i20910-Panama_Iowa.html
Another recent post may be useful to you. Jessica provided information about expat contacts in Panama City, and you may want to reach out to them. You can read her post here: http://internationalliving.com/forums/topic/expat-groups-or-meet-in-panama-city/
You’ll find excellent resources right here at your Panama Insider page about traveling around Panama, and these entries may help you plan your trip: http://internationalliving.com/country_clubs/panama/travel-itineraries/
Good luck with your planning and please post again if you have more questions.
Linda