First, I must say that I find it odd that the web-server is storing its data in text files and having customized algorithms to manipulate the data. Giving that lots of data manipulation are occurring, this seems highly inefficient to me; a database would be better in my opinion.
From your side however, it all depends on the data you're processing from the server. You can use the following:
SqlLite or CoreData - Using these approaches is one route to go. But you may have the extra work of sorting out the data into tables etc.. usual database stuff. Refer to the answer on this link to get a better idea on the difference between the SqlLite and CoreData or Google a bit.
Just save the data in files just like its being done on the server. There are several ways to do this including Property List files etc. Here's one method where it's being done using NSCoding.
I would say that the approach#1 approach has the advantage in organization and speed (has the advantage any relational database would have). However, the disadvantage is you will take more time properly setting up this correctly.
Approach#2 might save you time as you would not have to consider structure of data or anything, just dump and save. However, ask yourself, is the size of server data liable to increase? is speed a factor in processing? etc etc.. Think about long term, not just the short fix. This approach is right for you if the data will be relative the same and load is small.
The approach you choose largely depends on your data load and furture project scope (be it quick fix, or longer term improvement of current system) in my opinion.