According to Fundamental Theorem of Arithmetic any positive whole number is the product of primes. Therefore, I can create unique numbers by multiplying $n$ primes.
If I have a list of length $n$ generated by a finite sequence in the set ${1, -1}$, e.g., $(1, 1, -1, -1, 1, ...)$, and I do a dot product with a sequence of $n$ primes starting with 3, e.g., $(3, 5, 7, 11, 13, ...)$, do I have any guarantee that I will generate unique numbers by doing different permutations of 1s and -1s?
I know that I can't if my prime's sequence start with 2, $[1, 1, -1] \cdot [2, 3, 5] = [-1, -1, 1] \cdot [2, 3, 5]$, but I'm not sure for the cases where my sequence starts with higher primes.
No, there is no such guarantee. Some quick Python code shows that, for $[3,5,7,11,13]$ we have: $$7=3(1)+5(-1)+7(1)+11(-1)+13(1)$$ and $$7=3(-1)+5(1)+7(1)+11(1)+13(-1)$$ This is enough to disprove your claim, but this is just one counter-example, I found many others with my code.
Note: This is the smallest list for which a counter-example is found. For smaller lists, your claim holds.