Just to be clear, I never used disposable sippies if I was in any kind of position to be able to wash them; well, that is, when DS still used them as a regular thing. The no-spill kind were certainly preferable, and I used to fill them w/ ice and use them as icepacks in an insulated bottle bag. I only used coffeehouse cups when washing just was not feasible at all, but it was a useful trick.
I washed sippies all over WDW when DS still used them; I carried around a small travel toothbrush that had the space for toothpaste in the handle, and I used that to clean the valves and remove the milk scum. I also carried a little vial of dish soap, and would take the valve and drop it into the cup, add soap and hot water, then put the lid back on and shake it. After a rinse it was usually plenty clean at that point. The other thing I always carried around in those days was a couple of flat cloth diapers in a ziploc; lots of uses, but also great for drying out wet sippy cups.
I also used heavier wipe-clean bibs; DS just tore off the cheap kind. They were hard to find where I live, so I made a whole bunch of them by buying a yard of slipcover upholstery vinyl, then cutting them to size and sticking on velcro dots at the neck. I made them long enough to cover his lap, too.