In practice, HTTPS and SSH are commonly used in the HAMNET for administrative purposes (many devices do not offer anything else, even).
Source: HAMNET operator. Ask me anything!
WRT to your "administrative purposes" 97.219(d)(1) not only permits HTTPS and SSH, but requires it, in order to authenticate, presumably so unidentified or unauthorized people can't reconfigure your transmitter on top of the local air traffic control or something. Under 97.309(b) as long as you don't obscure the meaning you're all good. Obviously accessing administrative URLs is not an obscure meaning. Likewise tunneling a VPN trying to mimic administrative traffic to do illegal file downloading or something would be obscuring meaning.
That said, I don't think this was ever intended to connect directly to the internet in this manner.
As long as the control operator, or her employer are not making money on the advertisement displayed it would likely be A-OK.
It's perfectly legal or order pizza via autopatch over amateur radio, although impractical now that everyone owns a cell phone.
Maybe I am getting some specific local or machine rules confused with the FCC's rules.
"...snip... Calls to place an order for a commercial product may be made such as the proverbial call to the pizza restaurant to order food, but not calls to one's office to receive or to leave business messages since communications on behalf of ones employer are not permitted..." http://www.arrl.org/phone-patch-guidelines