Asking Politely

Am I using correct grammar in this sentence: "I ask that you please review and reply to me with updates."

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Looking at your sentence we could change a couple things, to make it a bit more natural. For example: I ask that you please review ________ (identify the object or thing to be reviewed) and reply with any updates. (We can remove "to me" as it is assumed that the person will reply to you specifically.

I ask that you please review the report and reply with any needed updates (…and send me any updates).

I noticed that you titled this question "asking politely", so if you would like some other ways to ask things, in a polite way, here are some examples;

Would you be so kind as to....
If you wouldn't mind ______ing...
Could you please...