Why is my Corgi puppy peeing while walking, especially indoors?

Poor Riley! Urinating while walking can sometimes be normal in males - I have seen some male dogs who seem like they just can't be bothered to stop and stand still when they need to pee. However, if he's dribbling urine passively and seems to be completely unaware of it, I would be concerned that he may have a bladder problem of some kind that is causing him to have this issue.

In such a young dog, he may have a birth defect called ectopic ureters (meaning that the tubes leading from his kidneys to his bladder insert int he wrong place) - this is more common in female dogs, but can sometimes be seen in males as well. He could also have a problem with his urethral sphincter, or an bladder infection causing him to have difficulty "holding it" in the house.

I would recommend taking Riley to the vet for a complete check-up when you are able to. He/she can do a urinalysis to check for evidence of infection, and can ask you some additional questions to help determine if he needs additional testing such as x-rays, abdominal ultrasound, etc. to check for other problems.