You should be able to make the halwa with soy or hemp milk. I think if you use saffron and cardamom it should mask any taste of the milk. In addition, you can use ricotta to finish it off.
Here is how you use ricotta in carrot halwa.
First cook down ricotta with some milk powder and butter until it's dry and crumbly.
You make the gajar halwa and if you make it correctly it will take a lot of elbow grease. Once you cook the halwa down and it has no water left, add this dry ricotta, butter and milk powder mixture to it and fry it a bit more.
Add some sliced nuts to it along with some golden raisins and serve warm.
I love gajar halwa, it is a great dessert for a cold night and goes well with some hot chai. A very typical Indian comfort dessert.