<p>Is there any persistence solution for Common Lisp, such as Elephant, that allows function persistence? Currently my app stores an identifier on the db and later searches in a function table which it is, but this method does not allow dynamically created functions to be stored.</p>

Functions are opaque objects, so you won't have much luck storing them in files or something like that.  You can, however, store lists and
compile
them upon retrieval from the database.
This won't help you store closures, of course.  That would entail storing the lexical environment along with the code, neither of which you have (portable) access to.  Code that you compile from stored lists will have to rely entirely on global data or data stored in the database.
By the way, note that you can
funcall
symbols, so you don't need a function table for global functions.