The successful testing of non-Abelian statistics not only serves as a milestone in fundamental physics but also provides a quantum-gate operation in topological quantum computation. An accurate and efficient readout scheme of a topological qubit is an essential step toward the experimental confirmation of non-Abelian statistics. In the current work, we propose a protocol to read out the quantum state of a Majorana vortex qubit on a topological superconductor island. The protocol consists of four Majorana zero modes trapped in spatially well-separated vortex cores on the two-dimensional surface of a Coulomb blockaded topological superconductor. Our proposed measurement is implemented by a pair of weakly coupled Majorana modes separately in touch with two normal-metal leads and the readout is realized by observing the conductance-peak location in terms of the gate voltage. Using this protocol, we can further test the non-Abelian statistics of Majorana zero modes in the two-dimensional platform. A successful readout of a Majorana qubit is a crucial step toward the future application of topological quantum computation. In addition, this Coulomb-blockaded setup can distinguish Majorana zero modes from Caroli-de Gennes-Matricon modes in vortex cores.