1. Atoms with no electron will have mass, because mass depends on number of Protons and Neutrons though it has no electron.
2. Atoms can exist without electrons, Matter is build out of neutrons, protons (+) and electrons (-). Matter becomes stable only if it is electrically neutral.
3. So atoms without electrons do exist and must have their own states (charged or uncharged transfered back and forth in their environment)
4. Example : He2+ . It has 2 protons and 2 neutrons but no electrons.