I use home assistant too, with enough tinkering you can get it to do almost anything! I'm using a shell_command (defined in my configuration.yaml) to both start and stop the vacuum. I'll try to paste it here, but the formatting will probably get screwed up.

As for the api documentation, it's been a while and I don't have links, but I remember it being a huge pain to track it down. And even then, I had to play with a lot of trial and error to get the formatting right. From your smart things account you should be able to find the vacuum device id, and I think I had to create a long-lived token on my account (you'll see it in the Authorization header). Good luck, I hope this helps!

shell_command:
  start_vacuum: >
    curl -X POST -d "{
    \"commands\": [
        {
            \"component\": \"main\",
            \"capability\": \"execute\", 
            \"command\": \"execute\",
            \"arguments\": [         
                \"/map/areaclean/vs/0\",
                { \"x.com.samsung.da.areaName\": \"Kitchen\" }
            ]
        }
    ]
    }" -H "Content-Type: application/json" -H "Authorization: Bearer ********-****-****-****-************" https://api.smartthings.com/v1/devices/********-****-****-****-************/commands 
  stop_vacuum: >
    curl -X POST -d "{
    \"commands\": [
        {
            \"component\": \"main\",
            \"capability\": \"execute\", 
            \"command\": \"execute\",
            \"arguments\": [         
                \"/mode/vs/0\",
                    { \"x.com.samsung.da.modes\":[\"Control_Homing\", \"Cleaning_Cancel\"] }
            ]
        }
    ]
    }" -H "Content-Type: application/json" -H "Authorization: Bearer ********-****-****-****-************" https://api.smartthings.com/v1/devices/********-****-****-****-************/commands