Cell entry triggers timestamp THAT DOES NOT CHANGE

The only way to do it without VBA is to use a circular reference.

First, your if E2="" is redundant because of the next if testing if e2 = "No order"
So you don't need to bother testing if e2 is "", because the ="No order" will already be False and return ""
So this is sufficient
=IF(E2="No order",NOW(),"")

Now, first thing is to set E2 to a state that makes that formula return "",
So clear contents of E2

Go to File - Options - Formulas
Check the box "Enable iterative calculation"
Click OK

Change your formula to this (where F2 is the cell you're putting the formula into)
=IF(E2="No order",IF(F2="",NOW(),F2),"")

Now when you change E2 to No Order, F2 will populate with Now, and won't change unless you remove No Order from E2.