Shuttle from Venice to the Dolomites

That is a pretty specialized request which I would guess few people on the forum would have any first hand experience. I have used this service to find group transfer quotes in Europe.
https://www.limousine-center.com/

I would start with your hotel or whatever lodging you have set up in the mountains. They have arrangements with services that may include local drivers with vans who will come to Venice to pick you up. (We do this with transport from Bolzano every year. The driver comes in from The Val Gardena.)

Where, exactly, is your destination?

Is there some reason you can’t take the train? What is your destination?

Check out:

https://www.suedtirol.info/en
They have an entire page on "getting there."

For the western side of the Dolomites I take the train to Bolzano, and local buses go everywhere from there. Cab rides to Val Gardena from Bolzano cost about 100 euros per car.

I will taking a bus from Venice directly to Cortina to start my trip this January. My hotel in Cortina sent this information.

thank you for your kind message,
We are glad to inform you from VCE or TFS airport there is a bus called Cortina Express , you could book directly online at the following link: http://www.cortinaexpress.it/?lang=en at the price of € 27,00 per person approx.
The second option available is private transfer at the cost of € 240,00 per way (maximum 4 people), € 280,00 per way (for a maximum of 6 people) also bookable through us.
Please let us know if we could provide further assistance,
Thanks and Best Regards,
Viviana-Reception

See you on the Sellaronda!

With 15 people, charter buses are often the most cost effective means of transport, and trains really don't go into the Dolomites proper. They go "close" to the Dolomites.