A first step would be for you to reiterate to them (copying the credit card company) in writing what you said above ie you consider that no contract was completed and ask them to confirm that (a) they confirm ALL on-line bookings by e-mail and (b) to ask the company to forward you the original e-mail confirmations they claim they sent confirming the purchase. These should be the orginal dated e-mails. Your note should forward the timed and dated copies of the ignored e-mails you say was sent asking for confirmation of the booking. If they can produce these confirmations then your case is weakened, if they cannot it might be a bit stronger.
There is a potential flawy implied in your posting. That is that nobody tried to telephone Opodo when the e-mail wasn't answered. Some people might see this as a bit odd.
Suggest you come back with their responses.
Good luck.
I will try all this. Yes, in hindsight, they should have asked me what to do at the time, not weeks later. But they are in their late 60s, so cannot blame them too much. I am surprised they managed to do all this online. Strange, you would think the older generation would use the phone, but hey...