You can use both EF and stored procedures together. Its not a either/or situation.

Have you considered rolling your own? It really depends on what your project needs are. If you are just storing information and not linking (making relational), then simple serialize/deserialize of JSON files can work very easily and would not have any outside dependencies. If you are needing relational, LocalDB (SQLExpress) would be a solid choice but does come with a size limit. I think its 10GB now. Used to be 4. I would not use Access.

I have a similar situation. Code is in classic ASP. Best approach is to get the rules as they SHOULD BE from the business owners if you can. If not, at least document the screen level functionality. When you feel comfortable, create a road map . Take one piece at a time. I would start by identifying what could be converted to a service prioritizing what is re-usable first. (Authentication/Authorization would be a good start) For ease of maintenance alone, I think a rewrite is in order. The longer the client waits, the larger the risk of something being deprecated in a future release.