When registering to vote in US elections
This is a loaded statement. The US, and specifically UOCAVA, covers federal elections and not state elections. The page and instructions you link to cover both. You do not need to provide an employer to register to vote in VA for federal elections.
The VA website provides a list of ways to be eligible for an absentee ballot for local elections. Most non-military expats fall under categories 6C (former VA resident) and 6D (employed citizen who was never a VA resident). As your wife is from VA, if she was a VA resident in the past, then she would qualify under 6c and does not need to provide the name of her employer (but does need to provide the last date of residency). If you have never been a VA resident, then you may need to qualify under 6D and provide the name of your employer.
The most likely intent of requiring the employer name is that VA has a burdensome state tax code requiring anyone domiciled in VA to file a state tax return. In other words, VA lets employed expats vote in local elections because they will then potentially have to pay state income tax. They presumably do not let unemployed expats vote since they cannot collection state income tax.