Political beliefs or views are not a "protected class" in US federal employment law. That said, most employers at least say that they support political participation, but once you start taking public actions it becomes a bit different.

If a person made a large contribution or gave other public support to an extremist candidate, an employer might not want any association with that.

If running for an office is going to take up a significant amount of one's time, such that it might impact availability for work during normal hours, an employer would care about that.

In this specfic case, if she was featuring "I work for Instacart" in any way in her campaign they might care about that.

If she was saying anything inflamatory or controversial, Instacart might not want to be associated with that.

So running for a nonpartisan office such as a local council or school board or similar would normally not be any issue. Anything where party politics feature prominently is going to be riskier.

Apparently.

It's a standard practice in suppressing democratic norms yes. The US has a long sordid history of the elites absolutely hating when commoners practice democratic beliefs.

Let's not forget that many of our "founding fathers" deeply hated democracy and when speaking of the "democracy problem" of the 1780s in America it was mostly referring to commoners in Massachusetts participating in civic duties.

This tradition has never went away and is still practiced today.