SQL Server® 2016, 2017 and 2019 Express full download – iTecNote

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SQL Server Express edition should be fine for this scenario. It has all the core features of the full product, but as you said you are only really using it for data storage and some SPs, then you will not need any of the additional functionality available in the other versions (ie. reporting and analysis services). There are some limitations to the express version (the biggest being that the maximum database size is 4GB), but they should not really affect you unless your are building a very busy ASP.Net application.

Some public-facing websites use SQL Server Express as the database server (DotNetKicks being the only one I can remember at the moment) without issue.