We will be visiting Jasper in early December.What is the best way to get from Jasper to Emerald Lake and then on to Banff, we will then be going on to Calgary. Any feedback and options would be a great help.

How are you arriving in Jasper? The Brewster bus counter is in the east half of the ViaRail station. There are several car rental agencies within walking distance of the ViaRail Station.
If Calgary is only for a flight out, consider taking the train either to Edmonton or west to Vancouver.

We are coming from Vancouver into Jasper by train and we will fly out of Calgary.

That's good if you don't have to pay a drop off fee for leaving your Jasper rental car at the Calgary airport.
When I arrive in Jasper by train I leave by train.
The other option is a Brewster bus tour.

There is no train from Jasper to Calgary. You can take a bus from Jasper to Banff and then on to Calgary (or other vehicle transport like an airporter van), but depending on how long you'll be staying in Banff and what you want to do while there, being without a vehicle of your own could prove troublesome.

@SD a bus from Jasper to Banff won't help getting to Emerald Lake tho. Unless there's interest in backtracking -maybe 1.5 hrs each way? depending on road conditions which is anyone's guess- to go from Banff to Emerald Lk. If the OP rents a vehicle from a Banff location, that avoids the drop off fees mentioned.
There is Brewster's for Banff to YYC/Calgary International, scheduling can be checked on their website. Happy travels, D63

There are a couple of problems here. First of all, the timing.

Early December would be in the middle of the winter season. If there is heavy snow fall, the Icefield Parkway maybe closed. Even if the snow fall is not heavy, unless you have the right equipment (snow tires plus chains) and you are experienced at winter driving conditions, it would be "white knuckle" driving, along a strecth of highway where there is no service available, not even a gas station. It could be a receipe for diaster.
That would be problem number one. It doesn't matter if you drive or take a bus when the road is closed.

Problem #2, I am not sure about the Emerald Lake destination unless the OP is going to stay at the Emerald Lake Lodge. Other than cross country skiing, there is not much going on at the Emerald Lake area at that time of the year besides a comfortable hotel for getaway and good food. Back tracking is a must, and there is no bus service there.

A better option would be take the train from Jasper to Banff, $120 one way, rent a car in Banff and drive to Emerald Lake. The distance needed to travel and the roads are much better than the drive from Jasper, plus there are some service, and many more vehicles on the road than driving down from Jasper, in case of emergencies.

Return the car to Banff, take the Calgary airport shuttle service from Banff to the Calgary airport.