If you have tatami on the floor, I can definitely understand the problem. There is no way (well, no easy way) to clean those.
We too, have a no-shoe policy carried over from spending 6 years in Japan (but hardwood & carpet, no tatami). What we do is to make it an option, when we have parties. We just let people know "where they can put their shoes if they want to take them off". Many people would prefer to take their shoes off, while others want to keep them on. We usually get between 50% & 75% with no shoes, & everyone is comfy.
When we used to have wall to wall carpet, we just steam cleaned the next morning. We invested $200 in a Bissel steam cleaner, and it sooo paid for itself in 2months after 3 parties, and more than made up for any kind of expense in taking the "nerves" away. Vomit, diarrhea, dog hair, grape jelly, red wine, you name it... it's been on our floor. And out came the steam cleaner and poof. All better. Exploding diapers, playdates, gravity checks, pets... no worries. It comes out. The other alternative is either hiring it done for about 80, or doing the rug doctor thing for 45-85 (it works okay, but not as well as a pro or a home steam cleaner, and you have to buy all the cleansing stuff that goes along with.