Correctly say "I will let you know with further updates"

Try this - "I will notify you of further updates.

This is common in business AmE.

More colloquial, and common in casual situations:

• I'll let you know if anything changes.

"I will update you as the situation changes"

"Updates to follow" (this one suggests "if anything worth updating occurs")

"Further updates will be forthcoming"

I think it would be better to say, "I'll keep you in the loop".

Keep someone in the loop - To furnish someone with sufficient relevant information and include them in the decision-making process.

It's like asking someone to keep you updated on things that are going on. So it will save you from any trouble of saying someone to keep them updated.