We present a novel automata-based approach to address linear temporal logic modulo theory (LTL_f^MT) as a specification language for data words. LTL_f^MT extends LTL_f by replacing atomic propositions with quantifier-free multi-sorted first-order formulas interpreted over arbitrary theories. While standard LTL_f is reduced to finite automata, we reduce LTL_f^MT to symbolic data-word automata (SDWAs), whose transitions are guarded by constraints from underlying theories. Both the satisfiability of LTL_f^MT and the emptiness of SDWAs are undecidable, but the latter can be reduced to a system of constrained Horn clauses, which are supported by efficient solvers and ongoing research efforts. We discuss multiple applications of our approach beyond satisfiability, including model checking and runtime monitoring. Finally, a set of empirical experiments shows that our approach to satisfiability works at least as well as a previous custom solution.