Perhaps it will later on - apps are still relatively new and not all stores have them - but in any case Primark does tend to do things a bit differently from other outlets. A big difference between them, all the other three stores you mentioned and in fact pretty much every high street clothing retailer you can think of, is that Primark doesn't have an online store at all. They do have a website as you can see here, but it is really to advertise or give you an idea of what their latest products are, help you find your nearest store etc. You can't buy online, and you can't use an app to help you locate products either.
This may change, but it may not - Primark tend to succeed by focusing on the high street, and on keeping their prices below those of the competition, and as this has worked well for them they may prefer not to direct their energies anywhere else in the future.