I tend to write, "say whatever they want", but I'm always tempted to write "say what ever they want". Is it acceptable to split the word in this context?
No it is not acceptable to split the word in your context:
Here, whatever is a relative pronoun and it is always written as one word. The only instance where one will find what ever is in the interrogative. For example:
In these cases, ever (an adverb) modifies the pronoun what and, as such, it cannot and should not be compounded with what.
However, one could also have whatever as an interrogative pronoun:
And, finally, whatever, as an adjective, can be used in the following manner:
At first glance, "what ever" looks weird in this context, because I parse it with the "ever" referring to "want", as just another, less stilted way of writing "what they ever (i.e., always) want".
The "ever" in "whatever" serves more for emphasis than for its literal meaning. For example, "what you want" refers to a single, definite thing, while "whatever you want" refers to a thing chosen from any number of indefinite things. "Whatever" implies uncertainty or arbitrariness, and thus cannot really ever be replaced with "what ever".
But constructions such as this with "what ever" in place of "whatever" are so archaic and peculiar that I imagine few people, upon seeing the former, would assume that it meant anything other than the latter.
So you can say whatever you want.
The answer seems clear. But I always have a tendency to want to split it up when I ask