The soap injector only works with a low pressure tip at the end of the wand. Somehow, it creates the siphon to pick up the soap solution.
If you have more than 25 gallons and/or 15 hours of paint spraying to complete, DO NOT waste your time/money on a Wagner Power Painter, $60 unit. An airless sprayer will put down a lot more paint much, much faster. Spend about 600 bucks, get a set-up (Graco is nice and I've used them for years) with a decent gun, a fair length hose and dimensions suitable for your storage requirements.
A roller with a somewhat thick nap and a brush (back-brushing - wiggling in the voids for best coverage) will do a better, longer lasting job on wicker than any sprayer alone. It takes time, diligence, endurance and patience to perform the roller/brush method, and the result may only endure a scant 50% longer, but at least there will be the illusion that you care. Other than that, spray the BeJesus out of it. Some people spray the ever-livin'-heck out of the substrate and have a team member come behind them with the back-brushing action. Have two brushes handy, because an airless sprayer will get ahead of the back-brushing in short order. With proper technique, a small roller can be used for back-brushing instead of a brush, if the paint isn't gobbed on too much.
Wear a mask! No matter the sprayer, when you're finished spraying, cleanliness is next... The Wagner Power Sprayer is harder to clean than an airless. Use a product called "Pump Armor" (or similar) for the final cycle during cleaning and storage.
Practice on some scrap, first. If you go with an airless, you'll be surprised at the output. It can push a piece of newspaper across the room with one blast, so have a good size practice piece (card board, scrap drywall/plywood, etc.)