I am putting in a new kitchen. The room used to be a living room. My question is when I put in the plumbing do I have to run a vent stack to the outside of my house or since the house already has a vent will that work? Also I have seen at the hardware store a vent the hooks up under the sink. Would that work for me?

The vent in the store is an air admittance valve (AAV) and they can be used for sinks but not all codes allow. Will you be submitting for a building permit, if so the inspector will advise you on the AAV. Every drain needs 2 things, a trap and venting.

I am putting in a new kitchen. The room used to be a living room. My question is when I put in the plumbing do I have to run a vent stack to the outside of my house or since the house already has a vent will that work? Also I have seen at the hardware store a vent the hooks up under the sink. Would that work for me?
Every fixture that has a trap MUST have a vent. Having said that,if your area allows AAV;s you may install one downstream from the trap, (see image) and be vented. Good luck, Tom

Check with your local Building Inspector. AAV are not allowed in my area.