Yes it is possible to color the font of individual words or characters differently in the same cell. And yes it can be done via a formula, but with the formula acting as the trigger, as for instance in a worksheet calculation event. You need VBA to do it.
All that said, here is a way to do it using a worksheet change event, in a range of cells from A1:B10. Yes I know I just said it's possible with a calculation event, but because I don't know the range and all the possibilities of words and colors you have in mind, I wanted to at least show you that it is possible to achieve what you want. All we'd need to modify this for your situation is to loop through the range of cells housing your VLOOKUP formula, triggered by the Worksheet_Calculate event.
Because you apparently have a number of VLOOKUP return possibilities and goodness knows how many colors in mind, here is a Select Case structure to get you started. I have an ulterior motive by suggesting the _Change trigger - - if you can avoid the loops which I think would be needed to evaluate all the cells housing the formula, then maybe you'll consider having those determining values ("Part special Product approved", etc.) displayed by manual or VBA entry, not by calculation return. That way you can pinpoint the affected cell(s) and not loop unnecessarily through others.
Right click on your sheet tab, left click on View Code, and paste this into the sheet module.
By the way, if you are wondering why I included two cases (Case "" and Case Else) to exit the sub, it's just to save you the trouble of setting this up for you to enter your own code at either or both of those instances.
Anyway, here's the code to get you started.
Private Sub Worksheet_Change(ByVal Target As Excel.Range)
If Intersect(Target, Range("A1:B10")) Is Nothing Or Target.Cells.Count > 1 Then Exit Sub
Target.Select
Select Case Target.Value
Case "Part approved"
'Enter code here depending on desired font color
Case "Product approved"
'Enter code here depending on desired font color
Case "Part approved and Product approved"
'Enter code here depending on desired font color
Case "Part rejected Product approved"
With ActiveCell
'Format all characters as red
.Font.ColorIndex = 3
'Format characters 1 thru 5 back to black
.Characters(1, ActiveCell.Characters.Count - 25).Font.ColorIndex = 1
'Reformat characters 14 thru 30 back to black
.Characters(14, ActiveCell.Characters.Count - 0).Font.ColorIndex = 1
End With
Case "Part special Product approved"
With ActiveCell
'Format all characters as blue
.Font.ColorIndex = 5
'Format characters 1 thru 5 back to black
.Characters(1, ActiveCell.Characters.Count - 24).Font.ColorIndex = 1
'Reformat characters 13 thru 29 back to black
.Characters(12, ActiveCell.Characters.Count - 0).Font.ColorIndex = 1
End With
Case ""
Exit Sub
Case Else
Exit Sub
End Select
End Sub
Hope this helps.