The contact you/he are suggesting here is not reasonable.
Firstly can you imagine having a webcam on in your house 24/7 so the ex can see what you're up to at any time (just coz she fancied a nose). That's basically what you're asking her to allow in her house. The fact that your ex and you think this is reasonable just shows how little consideration you have given to this or to his ex's feelings or entitlement to privacy.
Secondly the age of the children is not conducive to chatting, either by phone or skype and certainly not email. Are they profocient with a keyboard and do you seriously expect them to be able to log in, type a message to dad and send it without help from their mum!?
The fact you think they will sit and chat and that a set time will not cause issues when they may well have to be dragged to the phone/computer away from a game, tv etc is unrealistic. Your dh needs to be prepared for the children to not want to speak to him and not take it as a personal or blame his ex for that.
I have to persuade my ds that when he emails or writes to his Grammy than he needs to write about 3 different things. It's not often something he chooses to do, he has to be persuaded and actively encouraged. This is something I believe is important so I do it. He's 12 now and it's not got any easier. And I doubt in your case that his ex wants to waste her time doing this for him (because it is for him not the dc's he wants it).
You need to look at what other ways of staying in touch your dp can do that are child focused, age appropriate and don't expect a response or put presure of the dc to sit still chatting/typing and certainly not in keeping secrets from their mum.
So your dh could perhaps regularly write, perhaps with a short postcard or sending some stickers for them (every week or so), which is much more exciting to get than an email. He should pick fun pictures and easy language and keep it short. Bear in mind it will be his ex reading these to the dc more than likely.
He could also buy them note paper and stamps and put his address on the envelopes for them. Encourage them to write back or just to draw a picture for him and help them choose some funky pens or stickers to put on the envelopes (but expect the novelty to wear off quickly).
He could record some bedtime stories onto CD that he has read for them (again keep to short stories, not novels), or read to them on skype and do that at a time they will want to sit and listen (is in pj's and ready for bed.
But basically he needs to lower his expectation that they will want to do this or think it's their mother who is distracting them. And he also need to be sensitive to her if he wants her to encourage this kind of contact. They do have a life away from him and he can't expect to impinge on it just because it suits him.