Are there any ways to get Realm Gold without a credit card?

Their FAQ makes no mention of other ways:

What is Realm Gold?

Realm Gold is a currency used to purchase non-essential items such as clothing and accessory dyes (and cloth patterns), pets, elixirs, and dungeon keys. You cannot buy the Wine Cellar Incantation.

Each account starts with 100 gold for free (which is not renewed each time your character dies), you may purchase gold by standing on the gold coin icon located at each market place. Additional Realm Gold may be purchased with Credit Card, Mobile, and Paypal payments.

As far as I've seen anywhere, the answer is no. I don't know of anything that lets you get more gold for free, but you can buy some things with fame instead.

In the United States you can buy Steam Wallet Codes from GameStop using cold, hard cash ($20 or $50). This also works for Kongregate Kreds.

If you live elsewhere, you can also transfer money from your bank account to your Paypal account, no credit card required. This requires some paperwork and a few days to setup, however, so it might be easier to just get yourself a prepaid card off your local bank; for example, in Italy, the underaged can get a Postepay Junior card with parental consent; as far as I can tell that will work on the VISA circuit.

In general I would be wary of services that trade your time for "free money." For example, you can't do TrialPay offers if you're not of age anyway — which is likely the very reason you're trying to not use a credit card to begin with. According to Kongregate's Terms of Sale (emphasis mine),

You must have a valid credit card or other valid payment method accepted by the Billing Company (collectively, "Payment Method") and have reached the age of majority in the state in which you reside in order to purchase Virtual Items and to establish a payment account, which may not be transferred to or used by any third party. You represent and warrant to Kongregate that [...]

(e) you have reached the age of majority in the state in which you reside [...].

If payment cannot be charged to your Payment Method or your payment or charge is returned for any reason, including without limitation chargeback, Kongregate and the Billing Company reserve the right to either suspend or terminate your access to the payment account and/or your ability to purchase Virtual Items without notice.

The fact the underaged can't spend (or get) money on the Internet without parental consent is pretty much by design. It's also, sadly, a shortcoming of the "free to play" model.

If all else fails you can "pay with mobile" through the game's web interface. That'll still get you busted by your parents, mind you.