May I say "please join me with my family for a reception immediately following the final selection" or is there a better way to phrase the statement?
Why can't you use me and my family? If it is only that you have been taught it is more polite to put yourself last, then you can say my family and me.
With there is probably understandable, but stilted and unnatural. Unless you are not a native English speaker, people are likely to try and find some extra meaning in your use of the unnatural word with.