We investigate how the electron-vibron coupling influences electron transport via an anisotropic magnetic molecule, such as a single-molecule magnet (SMM) Fe4, by using a model Hamiltonian with parameter values obtained from density-functional theory (DFT). The magnetic anisotropy parameters, vibrational energies, and electron-vibron coupling strengths of the Fe4 are computed using DFT. A giant spin model is applied to the Fe4 with only two charge states, specifically a neutral state with a total spin S=5 and a singly charged state with S=9/2, which is consistent with our DFT result and experiments on Fe4 single-molecule transistors. In sequential electron tunneling, we find that the magnetic anisotropy gives rise to new features in the conductance peaks arising from vibrational excitations. In particular, the peak height shows a strong, unusual dependence on the direction as well as magnitude of applied B field. The magnetic anisotropy also introduces vibrational satellite peaks whose position and height are modified with the direction and magnitude of applied B field. Furthermore, when multiple vibrational modes with considerable electron-vibron coupling have energies close to one another, a low-bias current is suppressed, independently of gate voltage and applied B field, although that is not the case for a single mode with a similar electron-vibron coupling. In the former case, the conductance peaks reveal a stronger B-field dependence than in the latter case. The new features appear because the magnetic anisotropy barrier is of the same order of magnitude as the energies of vibrational modes with significant electron-vibron coupling. Our findings clearly show the interesting interplay between magnetic anisotropy and electron-vibron coupling in electron transport via the Fe4. Similar behavior can be observed in transport via other anisotropic magnetic molecules.