Dave,
At the stores where I was the Internet Director, I had ISMs that worked cradle to grave. The customer had to ask for an ISM to get credit but all calls from the website and third party providers went to the internet department. You don't have phones across the street?
Is it the philosophy, of that store, that emails are internet and calls are not? I much prefer a phone call and all of my descriptions had a strong call to action to "call now". My GM knew that I had the strongest people. I was with the ISMs and heard every call and wouldn't let a ISM take a call if he wasn't up to the task.
The phone calls that are going to the floor, who is monitoring them? Do you have a call measurement system? How strong is the floor? Are green peas getting those calls? Those calls are asking about cars that are on the internet. Do the floor guys know the source of those calls? Are they from the website, AutoTrader, Cars.com or are they print or electronic?
On the "sufficient back and forth", the desk manager is going to rule in favor of his floor guys whenever possible.
"hey I've got something different from so and so" That is classic evidence of the "disconnect". Most of the time, you don't hear that. The customer thinks that you are trying to take advantage and blow out. It doesn't just happen when they are sent a quote from your store. It could be from another or they could have pricing from edmunds or some other online source.
There are a couple of things that I would change on your website. On the VDP, in the area where the customer puts in the information, you have the dealership phone number in bold. I would remove the phone number from that section. It dominates that area and the number is prominent at the top of the page. You have a button for "Make An Offer". Personally, those were designed by someone that has never sold cars. I have never received a reasonable offer from a customer. It is usually fifty cents on the dollar. I would change that to "Get Your ePrice" and change the pop up box accordingly.