Is the ability to do depth first search a proof of Turing Completeness? Can we write a non-TC automaton that does DFS? As the title states. Let's say we have a set of inputs that define a tree structure. Is it possible to construct an automaton that can perform depth-first search on this data that is not Turing-complete? If so, can we state that the ability to do DFS a sufficient proof of Turing-completeness?

For the purpose of the question, you can have some leeway about how the data is represented. As long as you can come up with a combination of a tree representation and an automaton that can DFS it that is less than Turing-complete, that's good enough. If it simplifies anything, you can make assumptions about the nature of the tree (e.g. binary or not), but a more generalizable answer would be preferred.
Most functions you'll ever deal with in practice are primitive recursive, a model of computation that is properly less powerful than TMs.

More specifically, the problem from the question can be implemented on linear-bounded automata (LBA), a model of computation that is equivalent with context-sensitive grammars -- properly less powerful than TMs.