Would a referendum on third term abortion be legal in Virginia if the only options were “yes” or “no”?
A state government can call a referendum about anything it likes. Making it truly binding can be quite hard, as any legislature can always pass a law repealing anything it decided previously. However the California ballot initiative process provides a possible template. This is part of the California state constitution so the legislature cannot overturn the results.
In the case of abortion any such state law would run into the US constitution. The situation is complex and not clearly defined, but the basic principle is that states have to comply with the federal constitution, including any Supreme Court judgements about what the constitution means. Therefore a vote to ban 3rd trimester abortion completely could not be converted into a constitutional law (at least at present), but a vote to make it more readily available could be implemented.