In static electricity, if an object becomes charged, does it become ions?

They may form "ions."

You are correct that the act of becoming charged means you have more or fewer electrons than protons. However, whether that qualifies as an "ion" depends on the material. Some materials will have electrons that hover close enough to an individual molecule that we can say that molecule is "ionized." However some materials, like metals, permit that electron to float around quite a bit. It stops being meaningful to think of the electron as "associated" with any one atom, so thinking of them as ions starts to get a bit difficult. It becomes more meaningful to say the entire object is "charged."