Use array.find in each values until you find the answer.

const drawers = [
  {
    "name": "locations",
    "values": [
      {
        "value": "dana-point-ca",
        "label": "Dana Point, CA"
      },
      {
        "value": "bronx-new-york",
        "label": "Bronx, New York"
      },
      {
        "value": "new-york-ny",
        "label": "New York, NY"
      }
    ]
  },
  {
    "name": "programAreas",
    "values": [
      {
        "value": "coral-conservation",
        "label": "CORAL CONSERVATION"
      }
    ]
  }
]

function getLabel(x) {
  for (const nameValues of drawers) {
    const values = nameValues.values
    const item = values.find(v => v.value === x)
    if (item !== undefined) {
      return item.label
    }
  }  
}

getLabel("bronx-new-york")  // 'Bronx, New York'
getLabel("coral-conservation")  // 'CORAL CONSERVATION'
getLabel("Value that does not exist") // undefined

for(let i = 0 ; i < drawers.length ; i++){
    let lbl = drawers[i].values.find(o => o.label === "Bronx, New York").label;
    console.log(lbl)
}

There is a couple of ways to achieve that. As below, you will get all the possible results. However, you might need to deconstruct the arrays in order to get the strings.

const drawers = [
  {
    "name": "locations",
    "values": [
      {
        "value": "dana-point-ca",
        "label": "Dana Point, CA"
      },
      {
        "value": "bronx-new-york",
        "label": "Bronx, New York"
      },
      {
        "value": "new-york-ny",
        "label": "New York, NY"
      }
    ]
  },
  {
    "name": "programAreas",
    "values": [
      {
        "value": "coral-conservation",
        "label": "CORAL CONSERVATION"
      }
    ]
  }
]

const enteredValue = "bronx-new-york";
const resultArrays = []

const onSearchLocation = () => {
 drawers.find(location => {
    const labels = location.values.map((place => { 
      const array = [];
      if(place.value === enteredValue) {
          array.push(place.label);
      } 
      
      if(array.length > 0) {
          resultArrays.push(array);
      }
     }
   ))
  })}

onSearchLocation();

console.log(resultArrays);