You say that the procedure communicates with your web site. I assume that the web site is calling this stored procedure. Or is the stored procedure invoking a REST service?
You say that the stored procedure is there. When you get this error message, do you run
SELECT * FROM sys.procedures WHERE name = 'YourSP'
If it is there, what is the value of schema_id? Does it match the schema that the web site uses?
My guess is that there is something that changes this procedure, possibly moves it to a different schema. You could create a DDL trigger that logs all DDL events to a table, so see if you find something this way.