No, it would not be secure… especially not without specific precautions and protocol implementations – which random.org doesn't offer in the first place. Also, random.org is – as far as I know – not certified, which downgrades it from a “trust” point of view.
But when talking about such randomness providers, I always like to mention a more professional project too, because many people aren't aware of its existance: the NIST Randomness Beacon…
If you check the project homepage at https://beacon.nist.gov/home you'll notice that even that project clearly states:
WARNING: DO NOT USE BEACON GENERATED VALUES AS SECRET CRYPTOGRAPHIC KEYS.
And there's a good reason for that: they are still researching potential implementation options as well as potential security strengths and weaknesses of such a solution.
I wouldn't count on random.org having solved all the problems the NIST Randomness Beacon is still researching.
As you noticed yourself, random.org states something similar:
We should probably note that while fetching the numbers via secure HTTP would protect them from being observed while in transit, anyone genuinely concerned with security should not trust anyone else (including RANDOM.ORG) to generate their cryptographic keys.
In terms of security, I therefore would personally recommend to distrust data from services like random.org even more than the beacon data generated by the NIST Randomness Beacon project in its current research phase… and until further notice, you aren't supposed to trust the Beacon data either.
Wrapping it up: it would not be secure to use random numbers from services like random.org in a cryptographic solution. This is underlined by the individual services' statements.
There are ample well-vetted and cryptographically secure alternatives to the need of falling back on using such services. It would be smarter to use those than to trust a 3rd party that puts emphasis on the fact that you should not trust its data for crypto purposes…
Sites like random.org may have their place, but not in the realms of cryptography.