Can you legally bet online on somebody's behalf with their written consent?

The first hurdle that you have to clear is determining that online betting is legal at all. I can't bet online under state law, but other states allow it. If online betting is generally legal for you, you still may have to clear statutory hurdles regarding betting for another person, so again you have jurisdictional questions that arise (both where you are and where the betting website is).

If it is not prohibited to bet online and to do so "for another person", then you just have to read the ToS. Breach of the ToS is not itself a crime, but an act which is prohibited under the ToS might in fact be a crime (which is why it is not allowed under the ToS). It may well be a crime to use false credentials to log in to a website, since the site may be subject to "know your customer" laws. Such an action might in fact be the crime of fraud (lying about who you are so that they will let you play, when it is material that you aren't who you claim you are); or, it could fall under the Perjury Act. In the US, especially where a party is required by law to get certain information, there is usually a "perjury statement" which warns you that lying can result in a perjury prosecution. The criminal potential may also be there via the Money Laundering Act. On the other hand, from what I can make out of the Gambling Commission regulations, they only warn that underage gambling is an offense, and seem to put the due diligence burden on the casino.

Under the ToS, there is probably a clause or two to the effect that they will not pay out if you break the rules, so it's not just a matter of a "ban". This can include making it impossible for you to withdraw funds in the account.

If the problem is that your 110 year old aunt just cannot handle the internet and you want to help her to bet on the ponies, you can certainly ask the website for senior assistance. They may say "Sorry, we can't help", and you may have to shop around for a site that offers senior assistance. Or you can directly place the bet yourself, and work out an agreement between you and your aunt (by "can" I mean "you can consider doing this", I don't mean that there isn't a law against doing this in your jurisdiction).