In 2008, my father was awarded a judgment against an individual. The person has since moved out of Florida to Sunbury, PA with his wife and kids.

I want to know, if I were to file a "write of execution" in Florida, will the Sheriff in Subury enforce it?

No.
The judgment could, however, be registered in Pennsylvania and then enforced as though it were an instate judgment.

What AK lawyer is referring to is called domestication of foreign judgment. Most states with the exception of NY and a couple of others allow a out of state (foriegn) judgment to be domesticated to their state. Depending on what the rules of civil procedure are for your state it could take 20 to 30 days for the judgment to be real in that state with possibly a notice of domestication having to be sent to the debtor. Check your rules of civil procedure in Pennsylvania and see how you should proceed.

What AK lawyer is referring to is called domestication of foriegn judgment. ...

I suspect that the terminology is different in different states.

But "Between two different States in the United States, enforcement [of out-of-state judgments] is generally required under the "full faith and credit" [clause]... which compels a State to give another State's Judgment an effect as if it were local." Enforcement of foreign judgments - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Enforcement_of_foreign_judgments)

Article IV, Section 1 provides that:

"Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State. And the Congress may by general Laws prescribe the Manner in which such Acts, Records and Proceedings shall be proved, and the Effect thereof."