I want to move the kitchen sink to the other side of the kitchen (approx. 10 feet away). Can I just extend the pipe and rely on the existing vent?
You don't provide much context to this question, but the short answer is there's a maximum distance between the trap and the vent (should be specified by your local code). If the vent is too far away, the sink won't drain well.
As you can see, for a 2" pipe (typical), you can only go 5' (but of course: check your local building codes for specifics). 10' is definitely too far.
Remember also you need to maintain a slope on the waste line (typically 1/4" per foot).
If you can run the vent line over to the new location as well, it may be acceptable. A picture/diagram of what you're doing would help provide a better answer.