Fukuoka is actually very foreigner friendly and you can just walk around the Nakasu area (Fukuoka's red light district) right across Canal City and ask if shops accept foreigners.
If you want to do some research beforehand, you can use Yoasobi Heaven to find gaijin-friendly soaplands. Fukuoka gets a lot of foreigner traffic from Korea/China, so more than half of the soaplands in Nakasu accept foreigners and you have a lot of shops to choose from. All the shops that I've been to all have Korean + Chinese + English versions of the shop rules and the girls that I chatted with say that they serve more Korean/Chinese travelers than locals, lol.
For reservation, if you can speak Japanese, calling the shop is the main way to make a reservation with the girls. Alternatively, you can message the shop on LINE in either Japanese or English (it has a built-in translator) if they have a LINE number available but be warned, they may take several hours to reply your messages. The shop manager is usually busy entertaining calling customers, so if you decide to message them on LINE, be sure to make your booking earlier in the day. Lastly, there is also the option of messaging girls with their work SNS/Twitter accounts listed on their profiles and book directly with them, but YMMV on this option as I have seen some girls complain about clueless foreigners DMing them before
If you actually go down to the Nakasu area, right after the Minamishin Bridge, there is a free "Tourist information center" that you could go to and get recommendations on the shop to go to. I haven't been there since I prefer to do my own research, but if you don't have the time to do that you can attempt to get some recommendations there with the help of Google Translate.
Talking about this makes me want to book my next trip to Fukuoka now