I'm building an iPhone app that collects user's physiology data (e.g., timing of heart beats), which will be about 5-50mb per session. I'd like to allow users to upload these data with their ID and comments to Azure, but I'm not quite sure which products to use. I was looking at Azure Blob storage but it does not have write permission to public. Thanks for your help in advance.
I won't comment on any data privacy topics around this and just caution you to look into this, regardless of the technical solution.
To your question: It really depends on what kind of data formats you are talking about. If you want to upload things like images, then yes, Blob storage would be a good fit. You would probably have some back-end service in the cloud, that the app calls and requests a one-time write-access token to a blob storage account. you return that SAS token to the app and the app can then upload data into that blob storage.
https://learn.microsoft.com/en-us/azure/storage/common/storage-dotnet-shared-access-signature-part-1