Employing a combination of symmetry analysis, low-energy modeling, and ab initio simulations, we predict the presence of magnetic-field-induced Weyl points close to the Fermi level in CaKFe4As4. Depending on the relative strengths of the magnetic field and of the spin-orbit coupling, the Weyl fermions can carry a topological charge of +/- 1 or +/- 2, making CaKFe4As4 a rare realization of a double-Weyl semimetal. We further predict experimental manifestations of these Weyl points, both in bulk properties, such as the anomalous Hall effect, and in surface properties, such as the emergence of prominent Fermi arcs. Because CaKFe4As4 displays unconventional fully gapped superconductivity below 30 K, our findings open a route to investigate the interplay between superconductivity and Weyl fermions.