Can well-formed formulas in predicate logic for a given signature be recognized in LOGSPACE?
Yes, well-formed formulas in predicate logic for a given signature be recognized in LOGSPACE, at least for the two notations (grammars) given in the question!

The normal notation is indeed a visibly pushdown language, with $\Sigma_c=\{(\}$, $\Sigma_r=\{)\}$. When we encounter the symbol $($, then we push the number of still expected occurrences of of terms or proposition (and whether we expect terms or propositions) on the stack. When we encounter the symbol $)$, then we first check that exactly the expected number of terms or proposition occurred, then overwrite the counter by the value on the stack, and decrease it by one. The counter is finite, because its maximum value is the maximum arity of the signature, and zero is its minimal value, because we reject when the counter would become negative.



The given RPN grammar is also a visibly pushdown language, because the maximum arity of the given signature doesn't exceed two. You push on the stack the variables, pop from the stack for the binary function, the quantifier and the equality relation, and do nothing for the rest. What we actually push on the stack is the type (empty, term, or proposition) of the top element of the "real RPN" stack, before the new element is pushed. Convince yourself that we can always know this type (variable is not a type here).

Also RNP formulas for signatures with higher arity can be recognized in LOGSPACE, because they can be converted in constant time and space to an equivalent signature that whose arity doesn't exceed two, by using a pair constructor.