First, a lot of ABDL websites have a non-identifying name for the charge on your card, so if someone was just looking at it they probably wouldn't know unless they're ABDL too. Second, you used an email to sign up for this website right? Third, diapers are incontinence products, which is a very common solution to many medical problems. I understand not wanting your data everywhere and getting targeted adds, but as far as people (meaning strangers, like cashiers, mail carriers, random person you pass in the street) discovering them, it's a relatively normal thing that most people don't notice, or even become embarrassed themselves and completely ignore it. Diapers are meant to be discrete, as long as you don't purposefully go out of your way to make them obvious, like wearing thin, tight clothing, wearing multiple or super bulky ones, etc.
I can totally understand wanting to keep this a secret and leaving no trace, I worry about it too. I'm scared that my boss will find out and fire me and I'll never find a job again (very unrealistic and making mountains out of mole hills). But from the few weeks I've been on here I'm getting the impression and a lot of people who don't wear diapers don't notice and don't care. Although making a separate prepaid visa card sounds like a great way to buy them discretely. Good luck!