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:
In the United States, some more concrete thoughts :
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.
My favorite dashboard is Our World in Data: it has time series and maps and lets me choose which countries to graph. But it’s only showing confirmed cases, and doesn’t show any unit smaller than a country. Plus on the day I’m writing this, the US was missing yesterday’s day. Plague Plus is trying to do true prevalence estimates, but I don’t like their methodology, in part because it’s the same for every country.
Can I do better than this? Is something out there that shows city level data, with time series? If I must use confirmed cases, how can I translate those numbers into true prevalence estimates, given region-specific conditions?