Recent work has shown that polymeric catalysts can mimic some of the remarkable features of metalloenzymes by binding substrates in proximity to a bound metal center. We report here an unexpected role for the polymer: multivalent, reversible, and adaptive binding to protein surfaces allowing for accelerated catalytic modification of proteins. The catalysts studied are a group of copper-containing single-chain polymeric nanoparticles (CuI-SCNP) that exhibit enzyme-like catalysis of the copper-mediated azide-alkyne cycloaddition reaction. The CuI-SCNP use a previously observed "uptake mode", binding small-molecule alkynes and azides inside a water-soluble amphiphilic polymer and proximal to copper catalytic sites, but with unprecedented rates. Remarkably, a combined experimental and computational study shows that the same CuI-SCNP perform a more efficient click reaction on modified protein surfaces and cell surface glycans than do small-molecule catalysts. The catalysis occurs through an "attach mode" where the SCNPs reversibly bind protein surfaces through multiple hydrophobic and electrostatic contacts. The results more broadly point to a wider capability for polymeric catalysts as artificial metalloenzymes, especially as it relates to bioapplications.