Kinsa, a company that sells smart thermometers, has a dashboard that shows which regions of the US have an unusually high number of fevers. They have previously used these methods to track regional flu trends in the US. (FitBit has done something similar.)

I wrote a post here describing my attempt to turn their data into a rough estimate of the total number of coronavirus infections in the United States. Something similar could be done for smaller regions.

There's currently a Foretold community attempting to answer this question here, using both general Guesstimate models and human judgement taking into account the nuances of each country. We've hired some superforecasters from Good Judgement who will start working on it in a few days.

The Johns Hopkins Center for Systems Science in Engineering has time series data at the state and province level for some countries (US, China, Canada, Australia). They used to have county-level data for the US but no longer provide it. Unfortunately the case numbers are only confirmed + presumptive positive, so it's not everything you're asking for, but it seems like it gets close.

https://github.com/CSSEGISandData/COVID-19/tree/master/csse_covid_19_data/csse_covid_19_time_series

A few theoretical data sources that are pretty geographically specific:

• Hospital patient care records
• 911 calls
• Patient records from 911 calls
• 311 calls
• 911 detailed questions and answers

In the United States, some more concrete thoughts :

• 911 calls can be aggregated by asking my day job to integrate with your database.
• Detailed Q and A in 911 can be aggregated by the Q and A software vendor's partners (hi!) or themselves.
• Patient data from 911 is aggregated sporadically at the state level (NEMSIS) and my employer
• Hospital data is aggregated by ESSENCE (CDC and state level) with an unknown delay
• There are also other sporadic aggregators of hospital data.

If this is too much advertising, you can edit or reject this comment.

To summarize, your question is somewhat technically feasible. It's just a lot of work.