Providing this info for you, and anyone else interested.
Start here.
https://store.digitalriver.com/store/dndi/DisplayHelpPage
Click any of the links that regard your account (e.g. "My Order History" or "My Payment Information"). You'll be directed to a login page, if you aren't already logged into your account. (I've merged my account with a Wizards account, so I can log into either place--it's slightly more convenient to log into the Wizards account and then click the DDI page link, but I've just provided that to you anyway.)
Once you're logged in, click "My Order History." This should now bring up a list of your previous orders. Click on any of the links which describe a
completed purchase (none of the others will work for this). If you have a preference about what kind of purchase you'd like to make, select an order with the appropriate price tag (if there's one available). I, for example, have both the three-month $23.85 type and the one-month $9.95 type, so I can renew either of those types of subscription. If you only have one kind of subscription listed, I'm afraid that'll be the only thing you can renew (using this method).
From that individual order page, click the "Manage Subscription" link on the right-hand side. This will take you to your account subscription page. There should be a link on the left-hand side (again in a grey box) which says "Renew Now." Click it, and you'll be taken to a shopping cart with a new order for the kind of subscription you had purchased with the old order you looked at earlier, because you're renewing
that specific order. You can then click "Check Out."
I have not tested it beyond this point, as I am not currently interested in renewing my subscription--I do not know 100% for sure it will work, but I do not see any reason why it wouldn't. Digital River still exists as a payment processor, the site is still up and running, new order IDs are still being handed out for pending shopping carts (see below), and the DDI tools still know whether or not your account has a subscription. That would
seem to be all the necessary parts in place. Let me know if it actually lets you place the order. Somewhat to my surprise, the two times I tested this, I
did create new "orders" in my order history--they're just listed as "unsubmitted" because I cancelled them instead of proceeding to checkout. (It's now three times, because I did it again to make sure I was accurately indicating the steps you'd need to take.)