Yes, acids ionises. Also, they ionises step-wise. First step of ionisation:
H2SO4 + H2O ---> H3O+ + HSO4 - is an (almost) complete ionisation process. The second step is an incomplete one. Though for the purpose of the HSC, it can be assumed that H2SO4 will ionise two water molecules to H3O + per molecule of H2SO4.