In spin-based quantum-information-processing devices, the presence of control and detection circuitry can change the local environment of a spin by introducing strain and electric fields, altering its resonant frequencies. These resonance shifts can be large compared to intrinsic spin linewidths, and it is therefore important to study, understand, and model such effects in order to better predict device performance. We investigate a sample of bismuth donor spins implanted in a silicon chip, on top of which a superconducting aluminum microresonator is fabricated. The on-chip resonator provides two functions: it produces local strain in the silicon due to the larger thermal contraction of the aluminum, and it enables sensitive electron spin-resonance spectroscopy of donors close to the surface that experience this strain. Through finite-element strain simulations, we are able to reconstruct key features of our experiments, including the electron spin-resonance spectra. Our results are consistent with a recently observed mechanism for producing shifts of the hyperfine interaction for donors in silicon, which is linear with the hydrostatic component of an applied strain.