Validation for numeric data Javascript

You can do :

if(numbers % 2 == 0 && numbers !=0) ...
if(numbers % 2 != 0 && numbers !=0) ...

so that you don't do anything when numbers == 0;

It's a little strange to call your variable numbers instead of number.

your function should be,

function numberFunction(){

    do  
    {
        var numbers = prompt("Please enter numbers. Enter empty string to exit.");
        if(numbers !=0 && !isNaN(numbers)) 
            (numbers %2 == 0)? (evenarray.push(parseInt(numbers))) : (oddarray.push(parseInt(numbers)));
    }while(numbers !== "");

    for(var i = 0; i < evenarray.length; i++)
        sumEven += evenarray[i];

    for(var i = 0; i < oddarray.length; i++)
        sumOdd += oddarray[i];

    avgEven = sumEven / evenarray.length;
    avgOdd = sumOdd / oddarray.length;

document.getElementById("even").innerHTML = evenarray.toString();
document.getElementById("sumEvenTotal").innerHTML = sumEven.toString(); //displays sum of even numbers.
document.getElementById("averageOdd").innerHTML = avgOdd; //displays average of odd numbers.
document.getElementById("averageEven").innerHTML = avgEven; //diplays average of even numbers.
document.getElementById("odd").innerHTML = oddarray.toString(); //displays all odd numbers that were entered.
document.getElementById("sumOddTotal").innerHTML = sumOdd.toString();
}

As you already have other answers with solutions to your particular issue, I would suggest a different approach. Think of the data you're manipulating: an array. Try to solve the issue only with data, no user input, no DOM manipulation; just data. This helps to separate concerns, and make your code easier to understand.

Since we're working with arrays, we can make use of some of the built-in JavaScript methods that are present in modern browsers, such as filter and reduce. These methods are in a way, alternatives to for loops, with some pre-defined behavior, and a callback function.

Now, let's think of the steps involved in solving your problem.

1. Get numbers from the user. We can represent this data as an array, as you were already doing.
2. We want all odd numbers, their sum and average.
3. We want all even numbers, their sum and average.
4. We display the data to the user.

In this solution I'm assuming you already have an array with the data, and will be focusing on points 2 and 3. Remember, think of data, user interaction shouldn't be mixed with your data logic. Instead of asking the user for a number on each loop, you could ask the user for a list of numbers directly; you avoid multiple prompts this way, and it lets you separate data and interaction nicely. Ideally you'd validate all user input to match your requirements.

// Helpers to work with numbers
var odd = function(x) {
  return x % 2 === 0;
};

var even = function(x) {
  return x % 2 !== 0;
};

var add = function(x, y) {
  return x + y;
};

function solve(ns) {
  // Solve the problem
  // with odd or even numbers
  var result = function(fn) {
    var xs = ns.filter(fn); // odd or even
    var sum = xs.reduce(add);
    return {
      numbers: xs,
      sum: sum,
      average: sum / xs.length
    };
  };
  // Return an object
  // with odd and even results
  return {
    odd: result(odd),
    even: result(even)
  };
}

var numbers = [1,2,3,4]; // from user input
var result = solve(numbers);

console.log(result.odd);
//^ {numbers: [2,4], sum: 6, average: 3}

console.log(result.even);
//^ {numbers: [1,2], sum: 4, average: 2}