Sweep public data to OSM, county GIS tax records?

Can public data be conveniently swept to OSM, for instance http://map.co.door.wi.us/map/

If so, how?

You need to consult the US mapping community. In the UK the answer would be no, as:

  1. government data is not normally in the public domain so a compatible licence would have to be established;

  2. even if there a licence for the data, that might not extend to the map used to locate its position;

  3. there is generalised dislike of mechanical imports;

  4. in many cases what is already mapped from looking at the ground is more correct than local government provided information.

I believe the US federal level would probably put the data in the public domain, but I’m not sure the same applies to state and county level data.

The preferred method is to produce some sort of comparison map and ask people to use it to find where there is a difference and then check what is actually on the ground.

I can’t work out from your description or the map you are referencing what this data represents, but data on OSM needs to be stable over periods of years.