I ended up figuring it out thanks to some people over on reddit. You just need to add

"ContainerOverrides": {
  "Environment": [
    {
      "Name": "var1",
      "Value": "{% $states.input.value1 %}"
    },
    {
      "Name": "var2",
      "Value": "{% $states.input.value2 %}"
    }
  ]
}

to whichever container you are trying to get the variables into on the State Machine/Step Function definition, under the Arguments field.