Breakdown on what I learned:
The Moorings: They are still working out the bugs in their new system, but the woman in Florida who handles receiving the orders, Rachelle, is very sweet. You'll be talking to her if your cart won't check out. We think the issue with my checkout is that I'm using an Amex card to pay.
Bobby's Market: Download the order sheet, not the spreadsheet, fill it out, and email it to provisioning@bobbyssupermarket.com. They will email back confirming that they got your order and will communicate by email regarding totals and questions.
One Mart: You'll want to print out their spreadsheet, fill it out, and email it to orders@onemartfoods.com. I called and got that information.
Riteway: Order on their website. It works and they'll send you a confirmation email with a list for you to retain for your records.
Caribbean Cellars: Email your order to provisioning@caribbeancellars.com and they will respond. Do not submit it through their website, as the online ordering doesn't work and the contact form puts your list of purchases into one jumbled paragraph that they will email you about because it is unreadable.
Overall everyone was nice to deal with. I just wish I didn't waste so much time trying to get my online orders to work, since the sites for CC & OM don't work.
Hope this help someone in the future when traveling to the BVI.
Cheers & Happy Boating everyone!