I don't get how that would solve you problem at all. You can implement a bitcoin miner using functional code then just add an http client as a dependency for getting data to/from the blockchain.
You can't perform HTTP requests from a pure function without making it obvious in its signature that it does side effects. For example in a language like Haskell:
add : Int -> Int -> Int
add x y = x + y
There is no way a function like this can run a Bitcoin miner, all it can do is to return an `Int`. In order to do side effects, a function must return a special `IO` type that should then be returned from `main` (and only then these side effects would be performed).
unsafePerformIO exists, though. I suppose Safe Haskell might provide what you want, but (as someone mentioned elsewhere), the point of evaluating a pure function is to use its result somehow; a pure function that is supposed to look up a bitcoin address in a Map, but is changed to always return a constant one, could be just as bad of a backdoor.