Dare I spoil the secret that if you sign up for Prime, order something on Woot, and then cancel Prime, you can still get a full refund on the membership?
Put it in your cart on the website (whatever you prefer using), then place the order on the app. Might not work if there is a BOC or woot off. But it works for other things. I just did it for a $10 t-shirt. The 10% covers sales tax for me with a small amount leftover.
The short answer is that you can’t. The work-around would be to browse the shirts and add them to your cart, while signed in, from your browser and then go through checkout in the app. That should get you the 10% discount for Prime members. You can browse the whole shirt catalog on your phone in the browser, but you have to tell your browser to request the desktop version. I wish Woot would make the full catalog available on mobile, but you can’t reason with monkeys.