If it is anything it's for me. But it is not a common construction in this context; I have found just one example of it in published literature on google books:
I specifically authorize the release of information pertaining to Slate Canyon's detention hearing paperwork on CHILD, including contact information for me and CHILD. (source)
I did find one example of of me (here), but it seems that was not written by a native speaker.
What makes the for me sentence a bit more difficult to process than it should be is this: the for me construction can also be used in the sense of on my behalf. Here's an an example: Can you look up her contact information for me?
Thus, although for me is correct, it would be much better to rephrase the statement in any of the ways suggested by the other answers here. These include:
Please update the contact information you have for me.1 (I think this is closest to your original one.)
Please update my contact details in your contacts list.1
Please update the relevant contact information.2
1Suggested by Dissabled Dave.
2Suggested by D Manokhin.