We present the first empirical constraints on the polymer scale describing polymer quantized GWs propagating on a classical background. These constraints are determined from the polymer-induced deviation from the classically predicted propagation speed of GWs. We leverage posterior information on the propagation speed of GWs from two previously reported sources: 1) inter-detector arrival time delays for signals from the LIGO-Virgo Collaboration’s first gravitational-wave transient catalog, GWTC1, and 2) from arrival time delays between GW signal GW170817 and its associated gamma-ray burst GRB170817A. For pure-GW constraints, we find relatively uninformative combined constraints of ν = 0.96+0.15 −0.21 × 10−53 kg1/2 and µ = 0.94+0.75 −0.20 × 10−48 kg1/2 · s at the 90% credible level for the two polymer quantization schemes, where ν and µ refer to polymer parameters associated to the polymer quantization schemes of propagating gravitational degrees of freedom. For constraints from GW170817/GRB170817A, we report much more stringent constraints of νlow = 2.66+0.60 −0.10 × 10−56 , νhigh = 2.66+0.45 −0.10 × 10−56 and µlow = 2.84+0.64 −0.11 × 10−52 , µhigh = 2.76+0.46 −0.11 × 10−52 for both representations of polymer quantization and two choices of spin prior indicated by the subscript. Additionally, we explore the effect of varying the lag between emission of the GW and EM signals in the multimessenger case.