One of the simplest useful axiomatic system in mathematics is the Peano Axioms for the natural numbers. Notably it has only a single constant (the number zero).
One might suppose that one can do away with this single constant by considering an axiomatic framework for the integers which does not have a distinguished starting value. The usual construction for the integers, which goes via the natural numbers seem to deny that possibility. (Alternatively one might show that it has a unique model, is this possible)?
ZFC also has constants via the axiom of infinity - that there is an infinite set - an alternative way of seeing this, is that in topos theory, which is a generalised set theory, the replacement for the axiom of infinity is the natural numbers object which is the categorical equivalent of the peano axioms, so we're back to the first case mentioned above.
Now, a topos does not neccessarily have an natural numbers object, so perhaps that could count as such a theory, when considered foundationally; or since toposes are described via category theory, perhaps category theory when considered foundationally is such a theory. Is this in fact true, or is in fact a constant smuggled in via some other (sneaky) way?
Of course, there are theories without constants - just take any theory and drop any references to constants; we can even find useful examples - a heap which essentially models a group with its conjugation action (and is also, amusingly enough, a natural example of a mathematical structure with a ternary rather than binary operation); the question I'm after, though, is that it must be of foundational character - like ZFC, Category Theory or the Peano Axioms.