How do I sort a collection?

Collection is a rather wrong object for sorting.

The very point of a collection is to provide very fast access to a certain element identified by a key. How the items are stored internally should be irrelevant.

You might want to consider using arrays instead of collections if you actually need sorting.

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Other than that, yes, you can sort items in a collection.
You need to take any sorting algorithm available on the Internet (you can google inplementations in basically any language) and make a minor change where a swap occurs (other changes are unnecessary as vba collections, like arrays, can be accessed with indices). To swap two items in a collection, you need to remove them both from the collection and insert them back at the right positions (using the third or the forth parameter of the Add method).

You could use a ListView. Although it is a UI object, you can use its functionality. It supports sorting. You can store data in Listview.ListItems and then sort like this:

Dim lv As ListView
Set lv = New ListView

lv.ListItems.Add Text:="B"
lv.ListItems.Add Text:="A"

lv.SortKey = 0            ' sort based on each item's Text
lv.SortOrder = lvwAscending
lv.Sorted = True
MsgBox lv.ListItems(1)    ' returns "A"
MsgBox lv.ListItems(2)    ' returns "B"

The code below from this post uses a bubble sort

Sub SortCollection()

    Dim cFruit As Collection
    Dim vItm As Variant
    Dim i As Long, j As Long
    Dim vTemp As Variant

    Set cFruit = New Collection

    'fill the collection
    cFruit.Add "Mango", "Mango"
    cFruit.Add "Apple", "Apple"
    cFruit.Add "Peach", "Peach"
    cFruit.Add "Kiwi", "Kiwi"
    cFruit.Add "Lime", "Lime"

    'Two loops to bubble sort
    For i = 1 To cFruit.Count - 1
        For j = i + 1 To cFruit.Count
            If cFruit(i) > cFruit(j) Then
                'store the lesser item
                vTemp = cFruit(j)
                'remove the lesser item
                cFruit.Remove j
                're-add the lesser item before the
                'greater Item
                cFruit.Add vTemp, vTemp, i
            End If
        Next j
    Next i

    'Test it
    For Each vItm In cFruit
        Debug.Print vItm
    Next vItm

End Sub

Late to the game... here's an implementation of the MergeSort algorithm in VBA for both Arrays and Collections. I tested the performance of this implementation against the BubbleSort implementation in the accepted answer using randomly generated strings. The chart below summarizes the results, i.e. that you should not use BubbleSort to sort a VBA collection.

You can download the source code from my GitHub Repository or just copy/paste the source code below into the appropriate modules.

For a collection col, just call Collections.sort col.

Collections module

'Sorts the given collection using the Arrays.MergeSort algorithm.
' O(n log(n)) time
' O(n) space
Public Sub sort(col As collection, Optional ByRef c As IVariantComparator)
    Dim a() As Variant
    Dim b() As Variant
    a = Collections.ToArray(col)
    Arrays.sort a(), c
    Set col = Collections.FromArray(a())
End Sub

'Returns an array which exactly matches this collection.
' Note: This function is not safe for concurrent modification.
Public Function ToArray(col As collection) As Variant
    Dim a() As Variant
    ReDim a(0 To col.count)
    Dim i As Long
    For i = 0 To col.count - 1
        a(i) = col(i + 1)
    Next i
    ToArray = a()
End Function

'Returns a Collection which exactly matches the given Array
' Note: This function is not safe for concurrent modification.
Public Function FromArray(a() As Variant) As collection
    Dim col As collection
    Set col = New collection
    Dim element As Variant
    For Each element In a
        col.Add element
    Next element
    Set FromArray = col
End Function

Arrays module

    Option Compare Text
Option Explicit
Option Base 0

Private Const INSERTIONSORT_THRESHOLD As Long = 7

'Sorts the array using the MergeSort algorithm (follows the Java legacyMergesort algorithm
'O(n*log(n)) time; O(n) space
Public Sub sort(ByRef a() As Variant, Optional ByRef c As IVariantComparator)

    If c Is Nothing Then
        MergeSort copyOf(a), a, 0, length(a), 0, Factory.newNumericComparator
    Else
        MergeSort copyOf(a), a, 0, length(a), 0, c
    End If
End Sub


Private Sub MergeSort(ByRef src() As Variant, ByRef dest() As Variant, low As Long, high As Long, off As Long, ByRef c As IVariantComparator)
    Dim length As Long
    Dim destLow As Long
    Dim destHigh As Long
    Dim mid As Long
    Dim i As Long
    Dim p As Long
    Dim q As Long

    length = high - low

    ' insertion sort on small arrays
    If length < INSERTIONSORT_THRESHOLD Then
        i = low
        Dim j As Long
        Do While i < high
            j = i
            Do While True
                If (j <= low) Then
                    Exit Do
                End If
                If (c.compare(dest(j - 1), dest(j)) <= 0) Then
                    Exit Do
                End If
                swap dest, j, j - 1
                j = j - 1 'decrement j
            Loop
            i = i + 1 'increment i
        Loop
        Exit Sub
    End If

    'recursively sort halves of dest into src
    destLow = low
    destHigh = high
    low = low + off
    high = high + off
    mid = (low + high) / 2
    MergeSort dest, src, low, mid, -off, c
    MergeSort dest, src, mid, high, -off, c

    'if list is already sorted, we're done
    If c.compare(src(mid - 1), src(mid)) <= 0 Then
        copy src, low, dest, destLow, length - 1
        Exit Sub
    End If

    'merge sorted halves into dest
    i = destLow
    p = low
    q = mid
    Do While i < destHigh
        If (q >= high) Then
           dest(i) = src(p)
           p = p + 1
        Else
            'Otherwise, check if p<mid AND src(p) preceeds scr(q)
            'See description of following idom at: https://stackoverflow.com/a/3245183/3795219
            Select Case True
               Case p >= mid, c.compare(src(p), src(q)) > 0
                   dest(i) = src(q)
                   q = q + 1
               Case Else
                   dest(i) = src(p)
                   p = p + 1
            End Select
        End If

        i = i + 1
    Loop

End Sub

IVariantComparator class

Option Explicit

'The IVariantComparator provides a method, compare, that imposes a total ordering over a collection _
of variants. A class that implements IVariantComparator, called a Comparator, can be passed to the _
Arrays.sort and Collections.sort methods to precisely control the sort order of the elements.

'Compares two variants for their sort order. Returns -1 if v1 should be sorted ahead of v2; +1 if _
v2 should be sorted ahead of v1; and 0 if the two objects are of equal precedence. This function _
should exhibit several necessary behaviors: _
  1.) compare(x,y)=-(compare(y,x) for all x,y _
  2.) compare(x,y)>= 0 for all x,y _
  3.) compare(x,y)>=0 and compare(y,z)>=0 implies compare(x,z)>0 for all x,y,z
Public Function compare(ByRef v1 As Variant, ByRef v2 As Variant) As Long
End Function

If no IVariantComparator is provided to the sort methods, then the natural ordering is assumed. However, if you need to define a different sort order (e.g. reverse) or if you want to sort custom objects, you can implement the IVariantComparator interface. For example, to sort in reverse order, just create a class called CReverseComparator with the following code:

CReverseComparator class

Option Explicit

Implements IVariantComparator

Public Function IVariantComparator_compare(v1 As Variant, v2 As Variant) As Long
    IVariantComparator_compare = v2-v1
End Function

Then call the sort function as follows: Collections.sort col, New CReverseComparator

Bonus Material: For a visual comparison of the performance of different sorting algorithms check out https://www.toptal.com/developers/sorting-algorithms/