i have a couple of suggestions, which may require either a little $$$ or some clever scrounging for items others don't want.
1. the vacuum. if your carpeting under your table is low-pile, you could use a hokey to do the daily clean up, which can be stored on a hook on the back of a door. or, if your vacuum is huge, glorious, and covetable, maybe someone wants to trade with you for a smaller, more practical model?
2. your bedroom closet. important documents? put them in a big envelope or folder labelled "important documents" and stash that elsewhere. i know you said space in your desk is limited, but what about right under your desk, right behind it (between it and the wall), under your sofa, on top of your fridge, on top of a kitchen cupboard - or just upright against a side wall of your closet? they can't take up that much room. do the same for everything else that doesn't belong in your closet. use underbed storage, or box up out-of-season stuff, keeping the boxes on the floor of your closet. whatever is sitting on the floor of your closet right now, can sit on top of the box(es).
3. kids room. do they have furniture to store the toys in and they just aren't putting them away, or do the toys not have a place to go? again you can use the space under the bed for extra clothes, or for small toys, books, whatever. if you need storage furniture, open shelving is easiest for kids. you can probably find something at goodwill or salvation army, but i also think those wire cubes from target are practical and handy. it's $15 or $20 for a box, which makes a 2x3 shelf unit (so six cubes), and they're also good in closets. in my last apartment, i had them in my livingroom (holding craft supplies, magazines, etc), covered up with fabric.
4. your desk. what about a shelf or a little crate to store your books in, right above or right next to your desk? you can get brackets and a plank very cheaply, or you may be able to find a wooden crate for free (grocer, liquor store). coffee cups? pick one
! reuse, you don't even have to wash it for weeks at a time. (seriously, i never wash mine
we are also a family of four with two bedrooms and few closets, so you have my sympathy. we have an unfinished basement, though, for storage - i don't know what i would do without it. the truth is, i can sit here and tell you what to do, but i need major intervention myself! good luck and happy organizing.