You really should re-think applying via the inland process - you are not required to do that just because you want to be in Canada during PR processing. There is no advantage in it for US citizens . . . and it could end up taking a couple of
years for you to get PR that way if you just enter Canada as a visitor and don't get documented temporary status. In addition, when you file inland you forfeit your right to appeal a refusal, and you end up "stuck" in Canada because, if you leave and are not allowed to re-enter, you lose the inland application. You can still apply outland, even while staying in Canada as a visitor. The application will be processed through Buffalo - they are currently finalizing spousal and dependent child aps in 4-9 months . . . inland takes 12-18 months, but that's only if you have documented temporary status and there are no complications. If your case is not straight-forward, they transfer to a local CIC office and your application becomes subject to sometimes very lengthy timelines. Some local offices in areas with a high influx of immigrants are so backed up it can take in excess of a year before your application is even looked at.
Filing inland also will not get your kids authorized to go to school right away - that's not saying that the school won't enroll them . . . some Provinces have mandatory attendance laws and will enroll your kids regardless of your immigration status (or lack thereof) but that does not mean that Immgration Canada has authorized them to attend school and, if they want to play hardball with you, they will accuse you of "contravening" the IRPA by sending your children to school without getting Study Permits, and in a worst case scenario, that can make you ineligible for PR.
I would advise that your husband get enrollment letters for the children from the local school district. This should be relatively easy to do when he shows that he has married you. Then all you have to do is mail the enrollment letter(s),
with a SP application and the required fees, to Buffalo. They will make a decision on the SP application(s) and, if they're approved, they'll mail a certificate type thing to you that you have to take to the border to get the Study Permits. That will authorize the children to attend school while waiting for PR and you won't have to worry about whether or not you'll be in trouble with Immigration for complying with a mandatory attendance law and sending them to school without SPs.
Just be sure you include all the information requested in the SP application checklist, and provide proof of your marriage, financial support from your husband, and (probably) copies of the authorizations to immigrate (if applicable) that you may or may not have to provide with your PR application - depending on the custody arrangement you have with your children's father.
If you need more info, check out the
US2Canada website - it's designed specifically to support spousal applicants from the United States.
