Some sports will work out no matter what shoes you are wearing. Witness this woman juggling a soccer ball in platform heels.
KiSik Lee developed the KSL shot sequence. USA Archery has adopted it in modified form call the NTS. Whichever system you are following, one of the first steps is stance. Stance is concerned with where your feet are in relationship to the bow/target and how your weight is distributed across your feed. The preferred stance is 60% of your weight on your toes and 40% on your heels. This is nearly opposite of what most running shoes are set up for.
The preferred shoe for archers is more flat footed. If you were to set out looking for archery shoes, you would likely only find one brand — Archery Laces made by Flex Archery. These shoes are made in Spain. I’ll guess the low demand keeps prices pretty high.
If you aren’t interested in converting for European sizes or backorders, you may just be able to shop for something more accessible like any lo-fi or skate tennis shoe. Brands like Converse Chuck Taylor, Vans, Reef, or Toms may work out just fine for you. Switching to a simpler shoe will give your archer the ability to feel their feet and work on their stance, the important first step in their shot sequence.