Is there an iOS program that performs multivariate linear regression?
I don't think Wolfram Alpha's main app or their Linear Algebra Course Assistant are there yet with general multivariate linear regression but their email support is great and you might ask them which product of theirs might get that first. These do not run offline - so you may have to wait for the whole enchilada to arrive. 

In the mean time, you will need to use something like Matlab mobile to connect back to a computer that can actually run the numbers for you. This is a step better than just running screens as it lets you do input natively on iOS and shuttles forth and back the data to the main engine running remotely.

As you commented, you could get a matrix manipulation tool - both the Linear Algebra Course Assistant and a few other apps like matrix calculator plus will invert matrices for you.