How can I show array results using jQuery conditional statement?

I'm not entirely sure what question you're asking so I'll answer several things that are going on in your question.

If you want to know how to get the value from the select object, you can do that like this:

$("#category").val()

If you want to know how to detect the "All" choice, then you should change your HTML to this (set a value for the "All" item):

<select id="category" name="category">
     <option value="All" selected>All</option>
     <option value="Dealer">Wholesale &amp;Retail</option>
     <option value="Wholesale">Wholesale</option>
     <option value="Retail">Retail</option>
</select>

And you can check for it with a conditional like this:

if ($("#category").val() === "All") {
    // All is selected
}

I also see in your code that you have an error in your conditional. You can't compare to multiple things like this:

&&(locationData['category']=== 'Dealer' || 'Wholesale' ||'Retail')

You have to compare to each one separately:

&& (locationData['category']=== 'Dealer' || locationData['category'] === 'Wholesale' || locationData['category'] === 'Retail')

Or, using a temporary variable to hold the comparison value, your code could look like this:

if (settings.maxDistance && !firstRun) { 
    var item = locationData['category'];
    if((locationData['distance'] < maxDistance) && (item === 'Dealer' || item === 'Wholesale' || item === 'Retail')) {
        locationset[i] = locationData;
    } else {
        return;
    }
} else {
    locationset[i] = locationData;
}

If you do this a lot, sometimes, it's useful to make a static object that you can use as a lookup table:

var categories = {Dealer: true, Wholesale: true, Retail: true};
if (settings.maxDistance && !firstRun) { 
    if((locationData['distance'] < maxDistance) && (categories.hasOwnProperty(locationData['category'])) {
        locationset[i] = locationData;
    } else {
        return;
    }
} else {
    locationset[i] = locationData;
}