I want to replace the faucet for the kitchen sink. Ideally I would like dual handle, goose neck faucet and a sprayer. However, the hole where my sprayer would go contains the vent for my dishwasher and my disposal is also hooked up to this. Is this vent needed above the sink? Is there a way to move the vent under the sink if it is necessary?

Take the air gap out. Run dishwahser hose to directly to disposal, tie dishwasher hose to the underside of the countertop. The "high loop", tying the hose up tight to the counter top, takes the place of the air gap.

Let me enlarge on Harolds explanation. Yes, you may remove the counter top air gap,(see image) and install a high loop, (see image). High loops are all we ever install in my area with not the first customer complaint.
Here's how to run a high loop air gap.
Your drain hose must be looped as high under the counter as it will go. All that means is when you run the line, strap it up so the loop is above the inlet of the disposal or whatever you are draining into. If this is not done the water that you discharge from the sink will drain back into your dishwasher. Iuse a 3/4" copper pipe strap to secure but a nail and some wire will work also. Siunce the hose has been cut to install the counter top air gap you might have to puchase a longer washer hose. Good luck, Tom