I am trying to remodel my half bath to a full. Currently the toilet is about a foot off the main stack. I want to move it to about 4-5 feet away from the main stack. What is the maximum distance it can be away from the main stack before it needs it's own vent?

One of mine, 4" drain, is 8' from toilet to main stack. Your local code may require a clean out. Toilets pretty much vent themselves during a flush as the water swirls down and the down flow vents through the main stack.

You are safe with moving toilet 5' away from the stock

Milo