Here is a link to a video about how to get started with training your puppy to accept foot cleaning. Teaching skills before you are going to want to use them in real life is an important part of a successful training plan. I started teaching Martok to allow foot handling from a young age and started working on teaching acceptance of the foot cleaner while everything outside was still very frozen. This has given me multiple months to practice this skill before the thaw and needing to keep his feet clean.
If you are late to the game and it is already muddy, having an alternate plan is a good way to buy you time for training. An alternative that works in many houses is confinement to the entry way or a kennel for a short time after coming inside. This will help prevent tracking of mud through the house and the dirt can be brushed off when dry. Confinement to an entry can be accomplished using x-pens, closed doors, or baby gates. Providing a food puzzle with some kibble in it can be a great way to help your dog be comfortable while confined to an area away from you.
In the video, Martok already has the foundation skills of consent training. If he did not, I would have either had to teach him that skill first or approached training in a different way. Taking small steps and reinforcing each movement in the correct direction helps to build a powerful reinforcement history with the newly introduced foot washer. It is important when introducing new items to keep things fun and not forced. If I had forced his paw into the cup right away, I could have inadvertently caused him to panic and become afraid of the foot washer. It is easier to build a positive association to something that is neutral than it is to change an emotional association from a negative one to a positive one. It is always best to start your training at an easier point than you believe is necessary. Doing so will prevent you from potentially scaring your dog or puppy.
Happy training. If you need help, please feel free to reach out for one-on-one assistance.