M.B.
I would honestly suggest you ask a general question as to company policies on WFM. If there aren't any and others are not doing it, then you'll start out entirely on the wrong footing to ask for a BENEFIT that no one else has. My company is EXTREMELY flexible about telecommuting, however, it's up to each boss AND they would NEVER let someone new do so. I've been with the company 20 years, have an exceptional work record and primarily work with folks that are in different timezones, countries, etc. so I have the flexibility WFM to suit my schedule. I also considered taking another job 3 year ago and wanted to telecommute as the company had others in my position that did so. However, they had recently turned down a newer employee for that option, thus they wouldn't even consider that I could do so.
So, back to my initial point - if the company does not have a policy, you really need to consider not taking the job. If they do have a policy that allows folks to WFM, then I think your best bet is to take the job, work hard for 6 months and show them you are independent and can get your job done with little direction. Then, you can sit down with your boss at your 6 month review and see if they are amenable to a trial WFM period. I would also suggest you be very flexible for example and start with one afternoon a week and see how it goes. For example, if someone tries to reach you and you're not there, then that will look really bad. But, if you work very hard to show how responsive you can be, then things should work out.
Also, it seems you want to work from home when your kids are home, rather than working from home while they're in school. Most companies have a policy that requires you to have a caregiver in your home if kids are there. When my kids were younger, it was virtually impossible to WFM with them around - it's actually far worse for them to have you physically there, but not really available to them. It's also very hard to maintain a professional environment with kids in the home office environment. I have the same problem with my dogs when I WFM, but I can always give them a bone and put them in the backyard - I have to find anything that will keep my kids occupied like that).
I have to WFM a couple nites a week and my teens now understand that I will defer anything (except for emergencies) when I'm working. But, other than these critical deadlines, they can interrupt as they need to.