Robeez makes a few pair of "real" shoes in that size. Also, Zappos has some "real" shoes in that size but they are expensive. New Balance apparently makes some shoes in that size, but not sure if the sole is good for a new walker.

Teeny-tiny Chucks worked well for our tiny-footed DD.

My kids wore Robeez or Nikes. Nikes come in size 2. We got Roshes or velcro styles like this

https://www.nike.com/t/revolution-4-infant-toddler-shoe-v2TE9RzV/943304-601

Stride Rite starts at 3 for new walker shoes. Or leather coverings like Robeez are perfectly fine outside in short term.

They actually do recommend the softer leather soles at this age even for outside wear. I love Pediped, which has a more padded sole for this size range. Their Originals styles start with newborn sizing and they have stiffer soles available starting at size 4.

See Kai Runsstarts at size 3.
https://seekairun.com/all/first-walker/?limit=20&_bc_fsnf=1&Size=3

My DC is 15 months with tiny feet’s - like just smaller than 4 inches. Seems like that’s a size 2 - I have some size 3 mon petit soft leather shoes (a little big but not horrid with socks) but need something for walking outside. Seems like non-suede/leather bottom shoes don’t start till size 4?? Anyone have any recommendations for shoes for tiny feet?