Yes and no. To simplify matters, you need not just a quantum computer but such a machine with the right sort of qubits. The right sort of qubits being logical qubits, not physical, many of which are needed for error correction.
It's not clear whether we can currently create a machine with sufficient logical qubits to run Shor's algorithm in a meaningful way.
However, out of an abundance of caution, we're "starting" now (some designs have existed for longer but this (NIST PQC) is the first competition, which focuses minds) so as to have something ready when/if a quantum computer becomes a reality.
