My spouse took the kids and moved out, and now she's telling me that I can't see them. Can she do this?
No. She can move out, but without a court order in effect, your spouse doesn't have the right to control your access to your children. With no court order in place, you can legally take the children whenever and wherever you want, for as long as you want. You have as much right to spend time with them as your spouse does.
Don't let your spouse dictate or interfere with spending time with your kids. Doing so 'validates' the notion that he or she actually has this control, and tends to legitimize their efforts to keep you and your children apart. If there is no court order in effect, you are free to visit and/or pick the children up from school, take them to your home, feed them, care for them, etc, just as any parent would be.
If there is no court order or parenting plan in place and your soon-to-be-ex is already attempting to interfere with the relationship between you and the children, we suggest Hiring An Effective Attorney immediately and filing for sole custody. Your attorney may suggest you locate your children and take them to your home until this matter is sorted out.
Be aware that behavior like this now bodes very badly for the future- if your spouse is acting this way now, they'll only become more difficult as time goes by. They will use your child as a tool and a weapon against you. Your child will be hurt by their actions, but they won't care. The fact is, your having custody may be your child's only hope of having a normal, healthy life.
If your soon-to-be-ex will not support your relationship between you and your children, then it is crucial for you to obtain legal custody. This is the only way that both parents' relationship with the children will be assured.
Hopefully, you are the more reasonable parent and would not interfere with the other parent's relationship with the children. Since the same can't be said for them, it would follow that you will need to be the one in control of the custody arrangements.
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