C.J.
wow. i'm not really excited about the other responses. for one it's not just a phase to ride out. it's a combination of her not having the language skills to communicate effectively with you, plus the resulting frustration from not getting what she needs, which ultimately is a LOT of love. at that age, kids need to constantly have activities that help them bond with their family. without that, she's learning not to trust anyone, and will only get worse as she gets older.
(and I don't at all think that you're jealous of your husband for staying home, i think you just want things to click, and know your children are developing in a positive environment.)
i would suggest taping some episodes of super nanny, esp. ones showing older kids throwing tantrums and having your husband watch them. when he sees how bad the behavior will become if he doesn't give more love and attention now, maybe that will be incentive enough. he should also realize how much it will affect his wallet too when they become teens and demand everything because their emotional needs weren't met as they were developing.
you CANNOT spoil a child with too much love! (of course, you can with too many things.) her vomiting means she is THAT physically anxious about what she needs and it should be seen as serious. depending on HOW alone she is left regularly, children like this can be suffering from attachment disorder, and this can turn into horrible activities such as pyromania, and hurting others by age 4 or 5 and cutting themselves when they are in their teens (my partner wants me to say even worse possibilities, but i don't think things are that bad in your situation). it's situations like this that Attachment Parenting came out of. you can look at http://www.attachmentparenting.org/ideals.shtml if you like.
of course the 7 yr old can't be expected to play with her alone all day, but her dad should be coordinating activities where both can play together & he can give them each positive praise for playing well together. dad should be teaching the 7 yr old how to teach the 3 yr old. it will make the 7 yr old feel very accomplished when he gets praise from his dad and teach him responsibility, and will bolster his self-esteem tremendously. the 3 yr old will obviously benefit by having a brother as a role model instead of a sibling to be jealous of. and the dad will benefit because he will know he's doing a good job as a parent.
if the dad still won't change his behavior while you're away, you should make arrangements with a (consistent) friend or nanny to give the children the emotional attachment they need while you're gone.
and don't leave the 7 yr old out of any new positive attachment techniques either, just because he's doing fine. he probably has a different personality/temperament & may be able to mask his emotions more easily. i wish you all the best. peace!