Stateless pseudorandom/hash algorithm for ℤⁿ → [0,1)
Every hash function is stateless; that's part of the definition of what it means to be a hash function.

A simple approach is to take a representation of the input in binary, using any standard encoding for the integers and for $n$-vectors of integers, and then hash the result using your favorite hash function.  The output will be a binary string.  Treat it as the binary representation of a number in the range $[0,1)$ (put a decimal point before the start of it, and interpret it in base 2).  That's your output.

Then, choose a suitable hash function for hashing binary strings.  If you need it for cryptographic purposes, you could use SHA256.  If you need it for non-cryptographic purposes, there are many standard, fast hash functions; choose one that meets your requirements.  Since you didn't state any precise requirements in the question, we have no basis to select among any of the standard solutions.