Hi J. - I lived in Seattle until recently and we used cloth diapers for our son until he was about 18 months old (he still has a big belly and being up and running the diapers wouldn't stay on all that well after 18 months, even with toddler-sized diapers). We purchased our diapering supplies at a store called "Birth and Beyond" - it's located in Madison Valley (on Madison near the Arboretum). They have diapers and wraps and they also have diaper/wrap combo options too, and the staff is very helpful. I had trouble finding very many stores that sold cloth on the Seattle side, but there may be more options on the East Side. Birth and Beyond has good quality pre-fold diapers and a number of wrap options. I was very happy with what we purchased.
You won't likely be able to check some out to try, but you can buy maybe one wrap and three diapers to start, or 2-3 wrap/diaper combos. I ordered some wraps online based on user reviews but had better luck with the in-store variety. Every kid's body is different and the wraps all fit differently (and hold up to leaks differently). We ended up having to use a disposable at night, although I know some people do fine with a wool night wrap for example (ours leaked like crazy). You can probably try out the different wraps in the store over your daughter's clothes to see what fits best. My tip to prevent leaks is to also get a water-resistant wrap (some are coated on the inside with a plasticy substance and some are not). Our non-water-resistant ones always leaked (although as I said...my son wasn't delicate about his routine).
One thing to think about is whether you want to launder the diapers yourself. You have to do a pre-wash cycle on new diapers (get instructions at the store) and then there's a specific routine and limitations on the kind of detergent you can use...plus the mess of course. My husband washed them and we had a front-load machine which got them really clean, but I know some people don't have the time or energy (or stomach) to do it themselves. If you don't want to wash them, I'd suggest that you sign up for a service and try it out that way. Just find one that you can cancel any time. There are a number of eco-friendly diaper services in the Seattle area.