I am trying to have a Vlookup that pulls from another file. The issue is I
want the name of the file to be the value within a specific cell. For
instance, cell A1 has the value "File#1" (through a formula). I want the
vlookup to pull from File#1 without hard coding this into the formula.

Essentially I want my vlookup to be something like:
=VLOOKUP(A5,'[A1.xls]Sheet 1'!$A$1:$Z$10,2,FALSE)

(lookup cell A5 within the file named File#1, which is the value in cell A1)