Can a website access Apple Health data? Apps on the phone can. Is it possible to use that health data on a website? 

Reading the docs (and I don't understand them well yet), it seems like there could be an option if it was a clinic record? https://developer.apple.com/documentation/healthkit/samples/accessing_health_records
From the beginning of Accessing Health Records:


  HealthKit’s clinical record support lets you read Fast Healthcare Interoperability Resources (FHIR) from the HealthKit store. Users can download their FHIR records from supported healthcare institutions. The system then updates the records in the background on a regular basis. 


So it's the other way round: Users can (with means not described on that page) get their FHIR records and load them into HealthKit (the local store of Health data on your iDevice). The document describes how to access that data from an application.