Which is grammatically correct to say: "for anyone to claim" or " for anyone who claims"?

Both are correct..All depends on the contexts where they are used

I think that both are correct.the only difference is the place of the phrase.If it were at the end of a sentence,I would prefer to claim.If there is a completion,I will prefer who claims

The former focuses on the claim and the latter focuses on the person.
For e.g: For anyone to claim  (emphasis on how to get the object) For anyone who claims (emphasis on the person to do an act)

For anyone to claim - refers to an idea in general - use of the infinitive to claim 

For anyone who claims -refers to a specific case that is being discussed