Custom visual based on seat allocation

I got this to work in the attached file by just modifying the Vega spec you posted slightly. Note that I needed to add a "person" field in my example to effectively generate a row per seat

{
  "$schema": "https://vega.github.io/schema/vega/v5.json",
  "width": 550,
  "height": 300,
  "signals": [
    {
      "name": "dataLength",
      "update": "length(data('dataset'))"
    },
    {
      "name": "row0Radius",
      "value": 280
    },
    {
      "name": "row1Radius",
      "value": 260
    },
    {
      "name": "row2Radius",
      "value": 240
    },
    {
      "name": "row3Radius",
      "value": 220
    },
    {
      "name": "row4Radius",
      "value": 200
    },
    {
      "name": "row5Radius",
      "value": 180
    },
    {
      "name": "row6Radius",
      "value": 160
    },
    {
      "name": "row0Circ",
      "update": "PI*row0Radius"
    },
    {
      "name": "row1Circ",
      "update": "PI*row1Radius"
    },
    {
      "name": "row2Circ",
      "update": "PI*row2Radius"
    },
    {
      "name": "row3Circ",
      "update": "PI*row3Radius"
    },
    {
      "name": "row4Circ",
      "update": "PI*row4Radius"
    },
    {
      "name": "row5Circ",
      "update": "PI*row5Radius"
    },
    {
      "name": "row6Circ",
      "update": "PI*row6Radius"
    },
    {
      "name": "totalLength",
      "update": "row0Circ+row1Circ+row2Circ+row3Circ+row4Circ+row5Circ+row6Circ "
    }
  ],
  "data": [
    {
      "name": "dataset",
      "transform": [
        {
          "type": "project",
          "fields": ["Person", "Party"]
        },
        {
          "type": "window",
          "ops": ["row_number"],
          "fields": [null],
          "as": ["index"],
          "sort": {
            "field": "Party",
            "order": "descending"
          }
        }
      ]
    },
    {
      "name": "placement",
      "transform": [
        {
          "type": "sequence",
          "start": 1,
          "stop": {
            "signal": "dataLength+1"
          },
          "as": "index"
        },
        {
          "type": "formula",
          "as": "wholeCirc",
          "expr": "totalLength/dataLength"
        },
        {
          "type": "window",
          "ops": ["sum"],
          "fields": ["wholeCirc"],
          "as": ["cumWholeCirc"]
        },
        {
          "type": "formula",
          "as": "row",
          "expr": "datum.cumWholeCirc <row0Circ?0:datum.cumWholeCirc <row0Circ+row1Circ?1:datum.cumWholeCirc <row0Circ+row1Circ+row2Circ?2:datum.cumWholeCirc <row0Circ+row1Circ+row2Circ+row3Circ?3:datum.cumWholeCirc <row0Circ+row1Circ+row2Circ+row3Circ+row4Circ?4:datum.cumWholeCirc <row0Circ+row1Circ+row2Circ+row3Circ+row4Circ+row5Circ?5:6 "
        },
        {
          "type": "joinaggregate",
          "fields": ["Person"],
          "ops": ["count"],
          "groupby": ["row"],
          "as": ["rowCount"]
        },
        {
          "type": "formula",
          "as": "rowCirc",
          "expr": "datum.row==0?(row0Circ/(datum.rowCount-1)):datum.row==1?(row1Circ/(datum.rowCount-1)):datum.row==2?(row2Circ/(datum.rowCount-1)):datum.row==3?(row3Circ/(datum.rowCount-1)):datum.row==4?(row4Circ/(datum.rowCount-1)):datum.row==5?(row5Circ/(datum.rowCount-1)):datum.row==6?(row6Circ/(datum.rowCount-1)):0"
        },
        {
          "type": "window",
          "ops": ["sum"],
          "fields": ["rowCirc"],
          "groupby": ["row"],
          "sort": {
            "field": "index",
            "order": "descending"
          },
          "as": ["cumRowCirc"]
        },
        {
          "type": "formula",
          "as": "cumRowCircAct",
          "expr": "datum.cumRowCirc - datum.rowCirc "
        },
        {
          "type": "formula",
          "as": "theta",
          "expr": "datum.cumRowCircAct==0?0:datum.row==0?(datum.cumRowCircAct/row0Radius):datum.row==1?(datum.cumRowCircAct/row1Radius):datum.row==2?datum.cumRowCircAct/row2Radius:datum.row==3?datum.cumRowCircAct/row3Radius:datum.row==4?datum.cumRowCircAct/row4Radius:datum.row==5?datum.cumRowCircAct/row5Radius:datum.row==6?datum.cumRowCircAct/row6Radius:0"
        },
        {
          "type": "formula",
          "as": "x",
          "expr": "datum.row==0?row0Radius*cos(datum.theta):datum.row==1?row1Radius*cos(datum.theta):datum.row==2?row2Radius*cos(datum.theta):datum.row==3?row3Radius*cos(datum.theta):datum.row==4?row4Radius*cos(datum.theta):datum.row==5?row5Radius*cos(datum.theta):datum.row==6?row6Radius*cos(datum.theta):0"
        },
        {
          "type": "formula",
          "as": "y",
          "expr": "datum.row==0?row0Radius*sin(datum.theta):datum.row==1?row1Radius*sin(datum.theta):datum.row==2?row2Radius*sin(datum.theta):datum.row==3?row3Radius*sin(datum.theta):datum.row==4?row4Radius*sin(datum.theta):datum.row==5?row5Radius*sin(datum.theta):datum.row==6?row6Radius*sin(datum.theta):0"
        },
        {
          "type": "window",
          "sort": {
            "field": "theta",
            "order": "ascending"
          },
          "ops": ["row_number"],
          "fields": ["row_number"],
          "as": ["lookup"]
        },
        {
          "type": "lookup",
          "from": "dataset",
          "key": "index",
          "fields": ["lookup"],
          "values": ["Party"],
          "as": ["finalParty"]
        }
      ]
    }
  ],
  "scales": [
    {
      "name": "x",
      "type": "linear",
      "round": true,
      "nice": true,
      "zero": true,
      "domain": {
        "field": "x",
        "data": "placement"
      },
      "range": "width"
    },
    {
      "name": "y",
      "type": "linear",
      "round": true,
      "nice": true,
      "zero": true,
      "domain": {
        "field": "y",
        "data": "placement"
      },
      "range": "height"
    },
    {
      "name": "color",
      "type": "ordinal",
      "domain": {
        "data": "placement",
        "field": "finalParty"
      },
      "range": {"scheme": "pbiColorNominal"}
    }
  ],
  "marks": [
    {
      "name": "marks",
      "type": "symbol",
      "from": {"data": "placement"},
      "encode": {
        "update": {
          "x": {
            "scale": "x",
            "field": "x"
          },
          "y": {
            "scale": "y",
            "field": "y"
          },
          "shape": {"value": "circle"},
          "size": {"value": 130},
          "stroke": {
            "value": "#4682b4"
          },
          "tooltip": {
            "signal": "datum"
          },
          "fill": {
            "scale": "color",
            "field": "finalParty"
          }
        }
      }
    }
  ]
}

This is the updated Vega specification (note you need to change the setting in the Deneb visual to use the "Vega" provider instead of the default "Vega-Lite" provider)