There are many articles online on how to wash a car without a hose. All you need is either 2 or 3 buckets, a good wash sponge or mitt (or 2) and a small amount of car wash soap. In the first bucket put the car wash soap. Use as little as possible. Wash the car one section at a time. Rinse the car wash sponge in the rinse bucket, then while it is very wet go over the area you just washed to rinse it. Repeat till whole car is clean. Then, if any residue on the car, take a third bucket of clean water and a second wash sponge and rinse the whole car again from the 3rd bucket (Or second one emptied rinsed and refilled). Then dry using either a chamois, an Absorber artificial chamois or a water blade.
You do not need a hose to rinse, it is nice to have one, but it is not needed.
At a commercial car wash, especially the do-it-yourself ones, do not use the brush they all scratch. Any car wash that has anything that spins or rubs against your vehicle will scratch it. We have discussed that before. If you go to a do-it-yourself one with the spray gun, bring your wash mitt and wheel sponge. You do not need buckets or your own soap you are paying a lot to use them might as well get the full advantage of what you are paying for, and I have used them in the past the soap they have in the gun works just fine. Some do-it-yourself car washes have a sign that says do not use buckets.
Many here recommend using dish soap to wash cars that is a bad idea, it is very harsh and is designed to remove gunk from metal dishes etc. and can damage the clear coat. It is better to use a car washing product. You don't need a premium product, but a car washing specific product is gentler, less harsh and do work. Like I said $8 for 100 ounces, and you need very little even if handwashing without a power washer.
If you want to spend money, they are all kinds of ways to make the process easier or faster. For my own situation, I got the
Hydroshot Portable Power Cleaners | WORX and the soap attachment. This method needs more water, about 10 gallons for the source(I happened to have that special extra tall bucket), and I have a 5 gallon bucket for rinsing my wash sponge. I keep a very cheap large foam sponge for my wheels and tires. There are pressure washers at all price ranges and usually are used with a constant water supply. My neighbor used a bucket on a stand (he made) he installed a hose spigot to the bottom of the bucket with some plumbing parts. It was gravity fed, and he used 4 5 gallon buckets, the one with the spigot and 3 others to wash his cars.
You laughed at my suggestion to just use 2 buckets, but I have washed cars that way before, and it does work.