First things first. lets determine if in fact you do have that option on the car. Go to your Options Codes list which every car comes with. From the factory, there are two of them attached to the side wall of the spare tire well. That Options Codes list will give a code for every aspect of your car including every option is has. If it's not in the original factory location, the previous owner may have placed it in the cars maintenance folder or the owner's manual.
Those Options Codes can be decoded using this link.
VW Group Option Codes Decoder

There's sensors in each door handle and the early 08/09 A4/A5 had issues with them getting impregnated with water which would cause them to fail. The updated versions are completely sealed and can be sourced from China very cheaply using the oem part numbers from ECS.
You can also have concurrently a faulty door lock actuator. I believe the part number's suffix letter is "B" but I found the correct pin number type that used the same part number with the "A" suffix to look and work just as well and I got hat as well from China at less then 1/5 of the dealer cost.
This is what your Options Codes List should look something like this.
On my 2008 A5 this is my code and decoded info for my keyless entry.
4K7 = Central locking system "Keyless Entry" "two-stage unlocking"

 

One other quick question; is it possible the previous owner could have had the option for the advanced key entry disabled, and I would need to enable it once again? I saw the spot for the key to be inserted in the doors in the part where it is opened. Haven't read up on what that is for as yet, but will. It crossed my mind if that may have something to do with it.

 

Anything is possible but in something like this, a kind of doubt it. You have to ask yourself what would prompt him to do that in the first place.
Lets get that Options List of yours out and look to see if ever had it in the first place.

 

I'm sorry, I have replied twice now (checked the codes and it does have it) but for some reason my reply isn't being posted. My second attempt said the post needed to be approved by admin. That was last Saturday.

So I had checked the codes and I do have the option. Been busy and haven't remembered to plug in the battery but hopeful to get that done on the weekend.

 

BTW, @jschrauwen, THANK YOU! Your post with the info on where to find the codes, and what to look for as well as the website to decode them was very helpful, and very much appreciated! I wish my responses from last Saturday were posted, as I did thank you in them. I really appreciate your responses. I have the 4K7 code, which said keyless entry and two stage unlocking.

 

You're most welcome. You mentioned about plugging in the battery? Is that mean connecting the battery terminals to the battery or connecting a battery charger. If it's the former, was there a recent battery change before or during this door issue? If you were experiencing a dying or faulty battery, the car's computer can detect this and as part of a power conservation means, the car's ecu will start turning off or shutting down less critical electronic/electrical systems. Systems like the keyless entry and proximity sensors might very well have been turned off as part of the power saving measures the car's ecu will employ. If a new battery is installed, the MMI will want to see a new battery code. Just change one of the digits in the old battery code and you should be good.